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putnam_2014_a1
|
Require Import Reals Factorial Znumtheory Coquelicot.Derive.
Theorem putnam_2014_a1
(f : R -> R := fun x => (1 - x + x^2) * (exp x))
(hf : forall x : R, forall n : nat, ex_derive_n f n x)
(coeff : nat -> R := fun i => Derive_n f i 0 / INR (fact i))
(n : nat)
(hcoeff : coeff n <> 0)
: exists a b: Z, b <> 0%Z /\ coeff n = IZR a / IZR b /\ Zis_gcd a b 1 /\ (a = 1%Z \/ prime a).
Proof. Admitted.
|
import Mathlib
open Topology Filter
/--
Prove that every nonzero coefficient of the Taylor series of \[(1 - x + x^2)e^x\] about $x=0$ is a rational number whose numerator (in lowest terms) is either $1$ or a prime number.
-/
theorem putnam_2014_a1
(f : β β β)
(hf : β x : β, f x = (1 - x + x ^ 2) * Real.exp x)
(hfdiff : ContDiff β β€ f)
(c : β β β)
(hc : β k : β, c k = taylorCoeffWithin f k Set.univ 0)
: β k : β, c k β 0 β β q : β, c k = q β§ (q.num = 1 β¨ Prime q.num.natAbs) :=
sorry
|
putnam_1962_a6
|
Require Import Ensembles QArith.
Theorem putnam_1962_a6
(A : Ensemble Q)
(hSSadd : forall a b : Q, (A a /\ A b) -> A (a + b))
(hSSprod : forall a b : Q, (A a /\ A b) -> A (a * b))
(hSScond : forall r : Q, (A r \/ A (-r) \/ r = 0) /\ ~(A r \/ A (-r)) /\ ~(A r /\ r = 0) /\ ~(A (-r) /\ r = 0))
: A = (fun r : Q => r > 0).
Proof. Admitted.
|
import Mathlib
/--
Let $S$ be a set of rational numbers such that whenever $a$ and $b$ are members of $S$, so are $a+b$ and $ab$, and having the property that for every rational number $r$ exactly one of the following three statements is true: \[ r \in S, -r \in S, r = 0. \] Prove that $S$ is the set of all positive rational numbers.
-/
theorem putnam_1962_a6
(S : Set β)
(hSadd : β a β S, β b β S, a + b β S)
(hSprod : β a β S, β b β S, a * b β S)
(hScond : β r : β, (r β S β¨ -r β S β¨ r = 0) β§ Β¬(r β S β§ -r β S) β§ Β¬(r β S β§ r = 0) β§ Β¬(-r β S β§ r = 0))
: S = { r : β | r > 0 } :=
sorry
|
putnam_1962_b6
|
Require Import Reals Ensembles Coquelicot.Hierarchy Finite_sets.
Local Coercion INR : nat >-> R.
Theorem putnam_1962_b6
(n : nat)
(a b : nat -> R)
(xs : Ensemble R)
(f : R -> R := (fun x : R => sum_n (fun k : nat => a k * sin (k * x) + b k * cos (k * x)) n))
(hf1 : forall x : R, (0 <= x /\ x <= 2 * PI) -> abs (f x) <= 1)
(hxs : cardinal R xs (2 * n) /\ (forall x : R, xs x -> 0 <= x /\ x < 2 * PI))
(hfxs : forall x : R, xs x -> abs (f x) = 1)
: (~exists c : R, f = (fun x : R => c)) -> (exists a : R, f = (fun x : R => cos (n * x + a))).
Proof. Admitted.
|
import Mathlib
open Real
/--
Let \[ f(x) = \sum_{k=0}^n a_k \sin kx + b_k \cos kx, \] where $a_k$ and $b_k$ are constants. Show that, if $\lvert f(x) \rvert \le 1$ for $0 \le x \le 2 \pi$ and $\lvert f(x_i) \rvert = 1$ for $0 \le x_1 < x_2 < \cdots < x_{2n} < 2 \pi$, then $f(x) = \cos (nx + a)$ for some constant $a$.
-/
theorem putnam_1962_b6
(n : β)
(a b : β β β)
(xs : Set β)
(f : β β β)
(hf : f = fun x : β => β k β Finset.Icc 0 n, ((a k) * Real.sin (k * x) + (b k) * Real.cos (k * x)))
(hf1 : β x β Set.Icc 0 (2 * Ο), |f x| β€ 1)
(hxs : xs.ncard = 2 * n β§ xs β Set.Ico 0 (2 * Ο))
(hfxs : β x β xs, |f x| = 1)
: (Β¬β c : β, f = fun x : β => c) β β a : β, f = fun x : β => Real.cos (n * x + a) :=
sorry
|
putnam_1966_a4
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals sequences.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Theorem putnam_1966_a4
(a : nat -> int)
(ha1 : a 1%nat = 2)
(hai : forall n : nat, ge n 1 -> ((exists m : int, a n + 1 = m ^+ 2) /\ a (n.+1) = a n + 2) \/ ((~ exists m : int, a n + 1 = m^+ 2) /\ a (n.+1) = a n + 1))
(f : nat -> R := fun n => n%:R + (floor (@Num.sqrt R n%:R + 1/2))%:~R)
: forall n : nat, ge n 1 -> (a n)%:~R = f n.
Proof. Admitted.
|
import Mathlib
open Topology Filter
/--
Prove that the $n$th item in the ascending list of non-perfect-square positive integers equals $n + \{\sqrt{n}\}$, where $\{m\}$ denotes the closest integer to $m$.
-/
theorem putnam_1966_a4
(a : β β β€)
(ha1 : a 1 = 2)
(hai : β n β₯ 1, a (n + 1) = (if β m : β€, m^2 = a n + 1 = True then a n + 2 else a n + 1))
: β n β₯ 1, a n = n + round (Real.sqrt n) :=
sorry
|
putnam_2005_b2
|
Require Import Nat List Reals Coquelicot.Coquelicot.
Import ListNotations.
Definition putnam_2005_b2_solution (n: nat) (k: list nat) := (n, k) = (1%nat, [1%nat]) \/ (n, k) = (3%nat, [2%nat; 3%nat; 6%nat]) \/ (n, k) = (3%nat, [2%nat; 6%nat; 3%nat]) \/ (n, k) = (3%nat, [3%nat; 2%nat; 6%nat]) \/ (n, k) = (3%nat, [3%nat; 6%nat; 2%nat]) \/ (n, k) = (3%nat, [6%nat; 2%nat; 3%nat]) \/ (n, k) = (3%nat, [6%nat; 3%nat; 2%nat]) \/ (n, k) = (4%nat, [4%nat; 4%nat; 4%nat; 4%nat]).
Theorem putnam_2005_b2
: forall (n: nat) (k: list nat), length k = n -> ((gt n 0 /\ (forall i : nat, In i k -> gt i 0) /\ fold_left add k 0%nat = sub (mul 5 n) 4 /\ sum_n (fun m => 1 / INR (nth m k 0%nat)) (n - 1) = 1) <-> putnam_2005_b2_solution n k).
Proof. Admitted.
|
import Mathlib
open Nat Set
-- Note: uses β β β instead of Fin n β β
abbrev putnam_2005_b2_solution : Set (β Γ (β β β€)) := sorry
-- {(n, k) : β Γ (β β β€) | (n = 1 β§ k 0 = 1) β¨ (n = 3 β§ (k '' {0, 1, 2} = {2, 3, 6})) β¨ (n = 4 β§ (β i : Fin 4, k i = 4))}
/--
Find all positive integers $n,k_1,\dots,k_n$ such that $k_1+\cdots+k_n=5n-4$ and $\frac{1}{k_1}+\cdots+\frac{1}{k_n}=1$.
-/
theorem putnam_2005_b2
: {((n : β), (k : β β β€)) | (n > 0) β§ (β i β Finset.range n, k i > 0) β§ (β i β Finset.range n, k i = 5 * n - 4) β§ (β i : Finset.range n, (1 : β) / (k i) = 1)} = putnam_2005_b2_solution :=
sorry
|
putnam_1970_b5
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals exp sequences topology normedtype.
From mathcomp Require Import classical_sets.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
Variable R : realType.
Theorem putnam_1970_b5_solution
(ramp : int -> (R -> R) := fun (n : int) => (fun (x : R) => if x <= -n%:~R then -n%:~R else (if -n%:~R <= x <= n%:~R then x else n%:~R)))
(F : R -> R)
: continuous F <-> (forall n : nat, continuous (ramp (n%:Z) \o F)).
Proof. Admitted.
|
import Mathlib
open Metric Set EuclideanGeometry Filter Topology
/--
Let $u_n$ denote the function $u_n(x) = -n$ if $x \leq -n$, $x$ if $-n < x \leq n$, and $n$ otherwise. Let $F$ be a function on the reals. Show that $F$ is continuous if and only if $u_n \circ F$ is continuous for all natural numbers $n$.
-/
theorem putnam_1970_b5
(ramp : β€ β (β β β))
(ramp_def : ramp = fun (n : β€) => (fun (x : β) => if x β€ -n then (-n : β) else (if -n < x β§ x β€ n then x else (n : β))))
(F : β β β)
: Continuous F β (β n : β, Continuous ((ramp n) β F)) :=
sorry
|
putnam_1972_b6
|
Require Import Nat Reals. From Coquelicot Require Import Complex Hierarchy.
(* uses (nat -> nat) instead of ('I_k -> nat) *)
Theorem putnam_1972_b6
(k : nat)
(n : nat -> nat)
(zpoly : C -> C := fun z : C => (1 + sum_n (fun i => Cpow z (n i)) (k - 1))%C)
(hk : (k >= 1)%nat)
(hn : (forall i : nat, i <= k - 1 -> n i > 0)%nat)
(hn' : (forall i : nat, i <= k - 1 -> forall j : nat, j <= k - 1 -> i < j -> n i < n j)%nat)
: (forall z : C, Cmod z < ((sqrt 5 - 1) / 2) -> zpoly z <> 0).
Proof. Admitted.
|
import Mathlib
open EuclideanGeometry Filter Topology Set MeasureTheory Metric
/--
Let $n_1 < n_2 < \dots < n_k$ be a set of positive integers. Prove that the polynomail $1 + z^{n_1} + z^{n_2} + \dots + z^{n_k}$ has not roots inside the circle $|z| < (\frac{\sqrt{5}-1}{2}$.
-/
theorem putnam_1972_b6
(k : β)
(hk : k β₯ 1)
(n : Fin k β β€)
(hn : β i : Fin k, n i > 0)
(hn' : β i j : Fin k, i < j β n i < n j)
(zpoly : β β β)
(hzpoly : zpoly = fun z => 1 + β i : Fin k, z^(n i))
: β z : β, z β ball 0 ((Real.sqrt 5 - 1)/2) β zpoly z β 0 :=
sorry
|
putnam_2003_b4
|
Require Import Reals ZArith Coquelicot.Coquelicot.
Theorem putnam_2003_b4
(a b c d e: Z)
(r1 r2 r3 r4: R)
(ha : ~ Z.eq a 0)
: let a := IZR a in
let b := IZR b in
let c := IZR c in
let d := IZR d in
let e := IZR e in
(forall (z: R), a * z ^ 4 + b * z ^ 3 + c * z ^ 2 + d * z + e = a * (z - r1) * (z - r2) * (z - r3) * (z - r4)) ->
((exists (p q: Z), r1 + r2 = IZR p / IZR q) /\ r1 + r2 <> r3 + r4) -> (exists (p q: Z), r1 * r2 = IZR p / IZR q).
Proof. Admitted.
|
import Mathlib
open MvPolynomial Set Nat
/--
Let $f(z)=az^4+bz^3+cz^2+dz+e=a(z-r_1)(z-r_2)(z-r_3)(z-r_4)$ where $a,b,c,d,e$ are integers, $a \neq 0$. Show that if $r_1+r_2$ is a rational number and $r_1+r_2 \neq r_3+r_4$, then $r_1r_2$ is a rational number.
-/
theorem putnam_2003_b4
(f : β β β)
(a b c d e : β€)
(r1 r2 r3 r4 : β)
(ane0 : a β 0)
(hf1 : β z, f z = a * z ^ 4 + b * z ^ 3 + c * z ^ 2 + d * z + e)
(hf2 : β z, f z = a * (z - r1) * (z - r2) * (z - r3) * (z - r4)) :
(Β¬Irrational (r1 + r2) β§ r1 + r2 β r3 + r4) β Β¬Irrational (r1 * r2) :=
sorry
|
putnam_2005_a5
|
Require Import Reals Coquelicot.Coquelicot.
Definition putnam_2005_a5_solution := PI * ln 2 / 8.
Theorem putnam_2005_a5
: RInt (fun x => ln (x + 1) / (x ^ 2 + 1)) 0 1 = putnam_2005_a5_solution.
Proof. Admitted.
|
import Mathlib
open Nat Set
noncomputable abbrev putnam_2005_a5_solution : β := sorry
-- Real.pi * (Real.log 2) / 8
/--
Evaluate $\int_0^1 \frac{\ln(x+1)}{x^2+1}\,dx$.
-/
theorem putnam_2005_a5 :
β« x in (0:β)..1, (Real.log (x+1))/(x^2 + 1) = putnam_2005_a5_solution :=
sorry
|
putnam_2008_b4
|
Require Import Nat ZArith Reals Coquelicot.Coquelicot.
Theorem putnam_2008_b4
(p: nat)
(hp : Znumtheory.prime (Z.of_nat p))
(c: nat -> Z)
(n: nat)
(h : nat -> Z := fun x => floor (sum_n (fun i => IZR (c i) * INR (x ^ i)) n))
(hh : forall (i j: nat), i <> j /\ and (le 0 i) (le i (p ^ 2 - 1)) /\ and (le 0 j) (le j (p ^ 2 - 1)) -> Z.modulo (h i) (Z.of_nat (p ^ 2)) <> Z.modulo (h j) (Z.of_nat (p ^ 2)))
: (forall (i j: nat), i <> j /\ and (le 0 i) (le i (p ^ 3 - 1)) /\ and (le 0 j) (le j (p ^ 3 - 1)) -> Z.modulo (h i) (Z.of_nat (p ^ 3)) <> Z.modulo (h j) (Z.of_nat (p ^ 3))).
Proof. Admitted.
|
import Mathlib
open Filter Topology Set Nat
/--
Let $p$ be a prime number. Let $h(x)$ be a polynomial with integer coefficients such that $h(0), h(1), \dots, h(p^2-1)$ are distinct modulo $p^2$. Show that $h(0), h(1), \dots, h(p^3-1)$ are distinct modulo $p^3$.
-/
theorem putnam_2008_b4
(p : β)
(hp : Nat.Prime p)
(h : Polynomial β€)
(hh : β i j : Finset.range (p ^ 2), h.eval i β‘ h.eval j [ZMOD p ^ 2] β i = j)
: (β i j : Finset.range (p ^ 3), h.eval i β‘ h.eval j [ZMOD p ^ 3] β i = j) :=
sorry
|
putnam_1976_b6
|
From mathcomp Require Import all_algebra all_ssreflect.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope nat_scope.
Theorem putnam_1976_b6
(sigma : nat -> int := fun N => \sum_(d <- divisors N) d)
: forall N : nat, sigma N = (N.*2 + 1) -> (exists m : nat, odd m = true /\ N = m ^ 2).
Proof. Admitted.
|
import Mathlib
open Polynomial Filter Topology ProbabilityTheory MeasureTheory
/--
Let $\sigma(N)$ denote the sum of all positive integer divisors of $N$, including $1$ and $N$. Call a positive integer $N$ \textit{quasiperfect} if $\sigma(N) = 2N + 1$. Prove that every quasiperfect number is the square of an odd integer.
-/
theorem putnam_1976_b6
(Ο : β β β€)
(hΟ : Ο = fun N : β => β d β Nat.divisors N, (d : β€))
(quasiperfect : β β Prop)
(quasiperfect_def : β N, quasiperfect N β Ο N = 2*N + 1)
: β N : β, quasiperfect N β β m : β€, Odd m β§ m^2 = N :=
sorry
|
putnam_2007_b3
|
Require Import Reals Coquelicot.Coquelicot.
Definition putnam_2007_b3_solution := let a := (1 + sqrt 5) / 2 in (2 ^ 2006 / sqrt 5) * (a ^ 3997 - Rpower a (-3997)).
Theorem putnam_2007_b3
(X := fix x (n: nat) :=
match n with
| O => 1
| S n' => 3 * x n' + IZR (floor (x n' * sqrt 5))
end)
: X 2007%nat = putnam_2007_b3_solution.
Proof. Admitted.
|
import Mathlib
open Set Nat Function
noncomputable abbrev putnam_2007_b3_solution : β := sorry
-- (2 ^ 2006 / Real.sqrt 5) * (((1 + Real.sqrt 5) / 2) ^ 3997 - ((1 + Real.sqrt 5) / 2) ^ (-3997 : β€))
/--
Let $x_0 = 1$ and for $n \geq 0$, let $x_{n+1} = 3x_n + \lfloor x_n \sqrt{5} \rfloor$. In particular, $x_1 = 5$, $x_2 = 26$, $x_3 = 136$, $x_4 = 712$. Find a closed-form expression for $x_{2007}$. ($\lfloor a \rfloor$ means the largest integer $\leq a$.)
-/
theorem putnam_2007_b3
(x : β β β)
(hx0 : x 0 = 1)
(hx : β n : β, x (n + 1) = 3 * (x n) + β(x n) * Real.sqrt 5β)
: (x 2007 = putnam_2007_b3_solution) :=
sorry
|
putnam_2004_a3
|
Require Import Reals Coquelicot.Coquelicot.
Theorem putnam_2004_a3
(u : nat -> R)
(hubase : u O = 1 /\ u (S O) = 1 /\ u (S (S O)) = 1)
(hudet : forall n : nat, u n * u (Nat.add n 3) - u (Nat.add n 1) * u (Nat.add n 2) = INR (fact n))
: forall n : nat, exists m : Z, u n = IZR m.
Proof. Admitted.
|
import Mathlib
open Nat Topology Filter
/--
Define a sequence $\{u_n\}_{n=0}^\infty$ by $u_0=u_1=u_2=1$, and thereafter by the condition that $\det \begin{pmatrix}
u_n & u_{n+1} \\
u_{n+2} & u_{n+3}
\end{pmatrix} = n!$ for all $n \geq 0$. Show that $u_n$ is an integer for all $n$. (By convention, $0!=1$.)
-/
theorem putnam_2004_a3
(u : β β β)
(hubase : u 0 = 1 β§ u 1 = 1 β§ u 2 = 1)
(hudet : β n : β, Matrix.det (fun i j : Fin 2 => u (n + i * 2 + j)) = (n)!)
: β n : β, β m : β€, u n = m :=
sorry
|
putnam_1990_b3
|
Require Import Ensembles Finite_sets Reals Coquelicot.Coquelicot.
Open Scope R.
Theorem putnam_1990_b3
(E : Ensemble (matrix 2 2))
(hE : forall (A: matrix 2 2), E A ->
forall (i j: nat), and (le 0 i) (lt i 2) /\ and (le 0 j) (lt j 2) ->
(coeff_mat 0 A i j) <= 200 /\ exists (m: nat), coeff_mat 0 A i j = INR m ^ 2)
: (exists (sz : nat), gt sz 50387 /\ cardinal (matrix 2 2) E sz) -> exists (A B: matrix 2 2), E A /\ E B /\ A <> B /\ Mmult A B = Mmult B A.
Proof. Admitted.
|
import Mathlib
open Filter Topology Nat
/--
Let $S$ be a set of $2 \times 2$ integer matrices whose entries $a_{ij}$ (1) are all squares of integers, and, (2) satisfy $a_{ij} \leq 200$. Show that if $S$ has more than $50387$ ($=15^4-15^2-15+2$) elements, then it has two elements that commute.
-/
theorem putnam_1990_b3
(S : Set (Matrix (Fin 2) (Fin 2) β))
(hS : β A β S, β i j : Fin 2, (β x : β€, A i j = x ^ 2) β§ A i j β€ 200)
: (S.encard > 50387) β (β A β S, β B β S, A β B β§ A * B = B * A) :=
sorry
|
putnam_1992_a4
|
Require Import Nat Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1992_a4_solution (k : nat) := if odd k then 0 else pow (-1) (k/2) * INR (fact k).
Theorem putnam_1992_a4
(f : R -> R)
(hfdiff : forall k : nat, continuity (Derive_n f k) /\ forall x : R, ex_derive (Derive_n f k) x)
(hf : forall n : nat, gt n 0 -> f (1 / INR n) = (INR n)^2 / ((INR n)^2 + 1))
: forall (k : nat), gt k 0 -> (Derive_n f k) 0 = putnam_1992_a4_solution k.
Proof. Admitted.
|
import Mathlib
open Topology Filter Nat Function
abbrev putnam_1992_a4_solution : β β β := sorry
-- fun k β¦ ite (Even k) ((-1) ^ (k / 2) * factorial k) 0
/--
Let $f$ be an infinitely differentiable real-valued function defined on the real numbers. If
\[
f\left( \frac{1}{n} \right) = \frac{n^2}{n^2 + 1}, \qquad n = 1, 2, 3, \dots,
\]
compute the values of the derivatives $f^{(k)}(0), k = 1, 2, 3, \dots$.
-/
theorem putnam_1992_a4
(f : β β β)
(hfdiff : ContDiff β β€ f)
(hf : β n : β, n > 0 β f (1 / n) = n ^ 2 / (n ^ 2 + 1))
: (β k : β, k > 0 β iteratedDeriv k f 0 = putnam_1992_a4_solution k) :=
sorry
|
putnam_1990_b2
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals exp sequences normedtype topology.
From mathcomp Require Import classical_sets.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
Variable R : realType.
Theorem putnam_1990_b2
(x z : R)
(P : nat -> R)
(xlt1 : `| x | < 1)
(zgt1 : `| z | > 1)
(hP : forall j : nat, ge j 1 -> P j = (\prod_(0 <= i < j) (1 - z * x ^ i)) / (\prod_(1 <= i < j.+1) (z - x ^ i)))
: (fun n : nat => 1 + \sum_(1 <= j < n) ((1 + x ^ j) * P j)) @ \oo --> 0.
Proof. Admitted.
|
import Mathlib
open Filter Topology Nat
/--
Prove that for $|x|<1$, $|z|>1$, $1+\sum_{j=1}^\infty (1+x^j)P_j=0$, where $P_j$ is $\frac{(1-z)(1-zx)(1-zx^2) \cdots (1-zx^{j-1})}{(z-x)(z-x^2)(z-x^3) \cdots (z-x^j)}$.
-/
theorem putnam_1990_b2
(x z : β)
(P : β β β)
(xlt1 : |x| < 1)
(zgt1 : |z| > 1)
(hP : β j β₯ 1, P j = (β i : Fin j, (1 - z * x ^ (i : β))) / (β i : Set.Icc 1 j, (z - x ^ (i : β))))
: 1 + (β' j : Set.Ici 1, (1 + x ^ (j : β)) * P j) = 0 :=
sorry
|
putnam_2018_b6
|
Require Import Nat List Ensembles Finite_sets Reals.
Theorem putnam_2018_b6
(E: Ensemble (list nat) := fun l => length l = 2018 /\
forall (n: nat), (List.In n l) -> (n = 1 \/ n = 2 \/ n = 3 \/ n = 4 \/ n = 5 \/ n = 6 \/ n = 10) /\
fold_left Nat.add l 0 = 3860
)
(n : nat)
: cardinal (list nat) E n -> Rle (INR n) (Rmult (Rpower 2 3860) (Rpower (Rdiv (INR 2018) (INR 2048)) 2018)).
Proof. Admitted.
|
import Mathlib
/--
Let $S$ be the set of sequences of length $2018$ whose terms are in the set $\{1,2,3,4,5,6,10\}$ and sum to $3860$. Prove that the cardinality of $S$ is at most $2^{3860} \cdot \left(\frac{2018}{2048}\right)^{2018}$.
-/
theorem putnam_2018_b6
(S : Finset (Fin 2018 β β€))
(hS : S = {s : Fin 2018 β β€ | (β i : Fin 2018, s i β ({1, 2, 3, 4, 5, 6, 10} : Set β€)) β§ (β i : Fin 2018, s i) = 3860}) :
S.card β€ 2 ^ 3860 * ((2018 : β) / 2048) ^ 2018 :=
sorry
|
putnam_2009_a2
|
Require Import Reals Coquelicot.Coquelicot.
Definition putnam_2009_a2_solution : (R -> R) := fun x => Rpower 2 (-1 / 12) * Rpower (sin (6 * x + PI / 4) / (cos (6 * x + PI / 4)) ^ 2) (1 / 6).
Theorem putnam_2009_a2
(f g h: R -> R)
(a b: R)
(hab : a < 0 < b)
(hdiff : forall (x: R), a < x < b -> ex_derive f x /\ ex_derive g x /\ ex_derive h x)
(hf : forall (x: R), a < x < b -> Derive f x = 2 * (f x)^2 * (g x) * (h x) + 1 / ((g x) * (h x)) /\ f 0 = 1)
(hg : forall (x: R), a < x < b -> Derive g x = (f x) * (g x)^2 * (h x) + 4 / ((f x) * (h x)) /\ g 0 = 1)
(hh : forall (x: R), a < x < b -> Derive h x = 3 * (f x) * (g x) * (h x)^2 + 1 / ((f x) * (g x)) /\ h 0 = 1)
: exists (c d: R), c < 0 < d /\ (forall (x: R), c < x < d -> f x = putnam_2009_a2_solution x).
Proof. Admitted.
|
import Mathlib
open Topology MvPolynomial Filter Set
noncomputable abbrev putnam_2009_a2_solution : β β β := sorry
-- fun x β¦ 2 ^ (-(1 : β) / 12) * (Real.sin (6 * x + Real.pi / 4) / (Real.cos (6 * x + Real.pi / 4)) ^ 2) ^ ((1 : β) / 6)
/--
Functions $f,g,h$ are differentiable on some open interval around $0$
and satisfy the equations and initial conditions
\begin{gather*}
f' = 2f^2gh+\frac{1}{gh},\quad f(0)=1, \\
g'=fg^2h+\frac{4}{fh}, \quad g(0)=1, \\
h'=3fgh^2+\frac{1}{fg}, \quad h(0)=1.
\end{gather*}
Find an explicit formula for $f(x)$, valid in some open interval around $0$.
-/
theorem putnam_2009_a2
(f g h : β β β)
(a b : β)
(hab : 0 β Ioo a b)
(hdiff : DifferentiableOn β f (Ioo a b) β§ DifferentiableOn β g (Ioo a b) β§ DifferentiableOn β h (Ioo a b))
(hf : (β x β Ioo a b, deriv f x = 2 * (f x)^2 * (g x) * (h x) + 1 / ((g x) * (h x))) β§ f 0 = 1)
(hg : (β x β Ioo a b, deriv g x = (f x) * (g x)^2 * (h x) + 4 / ((f x) * (h x))) β§ g 0 = 1)
(hh : (β x β Ioo a b, deriv h x = 3 * (f x) * (g x) * (h x)^2 + 1 / ((f x) * (g x))) β§ h 0 = 1)
: (β c d : β, 0 β Ioo c d β§ β x β Ioo c d, f x = putnam_2009_a2_solution x) :=
sorry
|
putnam_2008_b5
|
Require Import Reals QArith Coquelicot.Coquelicot. From mathcomp Require Import div.
Open Scope R.
Definition putnam_2008_b5_solution (f : R -> R) := exists n : Z, f = (fun x => x + IZR n) \/ f = (fun x => -x + IZR n).
Theorem putnam_2008_b5
(f: R -> R)
(hf : (forall x: R, ex_derive f x) /\ continuity (Derive f))
: (forall (q: Q), exists (n1 n2 d: nat), Q2R q = INR n1 / INR d /\ f (Q2R q) = INR n2 / INR d /\ coprime n1 d = true /\ coprime n2 d = true) <-> putnam_2008_b5_solution f.
Proof. Admitted.
|
import Mathlib
open Filter Topology Set Nat
abbrev putnam_2008_b5_solution : Set (β β β) := sorry
-- {fun (x : β) => x + n | n : β€} βͺ {fun (x : β) => -x + n | n : β€}
/--
Find all continuously differentiable functions f : \mathbb{R} \to \mathbb{R} such that for every rational number $q$, the number $f(q)$ is rational and has the same denominator as $q$.
-/
theorem putnam_2008_b5
(fqsat : (β β β) β β β Prop)
(hfqsat : β f q, fqsat f q β ContDiff β 1 f β§ (β p : β, p = f q β§ p.den = q.den))
: β f : (β β β), (β q : β, fqsat f q) β f β putnam_2008_b5_solution :=
sorry
|
putnam_2010_a2
|
Require Import Reals Coquelicot.Coquelicot.
Definition putnam_2010_a2_solution (f: R -> R) := exists (c d: R), f = (fun x => c * x + d).
Theorem putnam_2010_a2
(f: R -> R)
: (forall (x: R) (n: nat), gt n 0 -> ex_derive f x /\ Derive f x = (f (x + (INR n)) - f x) / (INR n)) <-> putnam_2010_a2_solution f.
Proof. Admitted.
|
import Mathlib
abbrev putnam_2010_a2_solution : Set (β β β) := sorry
-- {f : β β β | β c d : β, β x : β, f x = c*x + d}
/--
Find all differentiable functions $f:\mathbb{R} \to \mathbb{R}$ such that
\[
f'(x) = \frac{f(x+n)-f(x)}{n}
\]
for all real numbers $x$ and all positive integers $n$.
-/
theorem putnam_2010_a2
: {f : β β β | Differentiable β f β§
β x : β, β n : β€, n > 0 β deriv f x = (f (x + n) - f x)/n} = putnam_2010_a2_solution :=
sorry
|
putnam_1992_a2
|
Require Import Reals Binomial Factorial Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1992_a2_solution := 1992.
Theorem putnam_1992_a2
(C : R -> R := fun a => (Derive_n (fun x => Rpower (1 + x) a) 1992) 0 / INR (fact 1992))
: RInt (fun y => C (-y - 1) * sum_n_m (fun k => 1 / (y + INR k)) 1 1992) 0 1 = putnam_1992_a2_solution.
Proof. Admitted.
|
import Mathlib
open Topology Filter
abbrev putnam_1992_a2_solution : β := sorry
-- 1992
/--
Define $C(\alpha)$ to be the coefficient of $x^{1992}$ in the power series about $x=0$ of $(1 + x)^\alpha$. Evaluate
\[
\int_0^1 \left( C(-y-1) \sum_{k=1}^{1992} \frac{1}{y+k} \right)\,dy.
\]
-/
theorem putnam_1992_a2
(C : β β β)
(hC : C = fun Ξ± β¦ taylorCoeffWithin (fun x β¦ (1 + x) ^ Ξ±) 1992 Set.univ 0)
: (β« y in (0)..1, C (-y - 1) * β k β Finset.Icc (1 : β) 1992, 1 / (y + k) = putnam_1992_a2_solution) :=
sorry
|
putnam_1969_b3
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals exp sequences topology normedtype trigo.
From mathcomp Require Import classical_sets.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
Variable R : realType.
Theorem putnam_1969_b3
(T : nat -> R)
(hT1 : forall n : nat, ge n 1 -> (T n) * (T (n.+1)) = n%:R)
(hT2 : (fun n => (T n)/(T n.+1)) @ \oo --> 1)
: pi * (T 1%nat) ^+ 2 = 2.
Proof. Admitted.
|
import Mathlib
open Matrix Filter Topology Set Nat
/--
Suppose $T$ is a sequence which satisfies $T_n * T_{n+1} = n$ whenever $n \geq 1$, and also $\lim_{n \to \infty} \frac{T_n}{T_{n+1}} = 1. Show that $\pi * T_1^2 = 2$.
-/
theorem putnam_1969_b3
(T : β β β)
(hT1 : β n : β, n β₯ 1 β (T n) * (T (n + 1)) = n)
(hT2 : Tendsto (fun n => (T n)/(T (n + 1))) atTop (π 1))
: Real.pi * (T 1)^2 = 2 :=
sorry
|
putnam_2022_a1
|
Require Import Reals Coquelicot.Coquelicot.
Definition putnam_2022_a1_solution : R -> R -> Prop := fun a b => (a = 0 /\ b = 0) \/ (Rabs a >= 1) \/ (0 < Rabs a < 1 /\ (b < ln (1 + ((1 - sqrt (1 - a ^2))/ a) ^ 2) - a * (1 - sqrt (1 - a ^ 2) / a) \/ b > ln (1 + ((1 + sqrt (1 - a ^ 2)) / a) ^ 2) - a * (1 + sqrt (1 - a ^ 2) / a))).
Theorem putnam_2022_a1
: forall a b : R, (exists! (x: R), a * x + b = ln (1 + x ^ 2) / ln 10) <-> putnam_2022_a1_solution a b.
Proof. Admitted.
|
import Mathlib
open Polynomial
abbrev putnam_2022_a1_solution : Set (β Γ β) := sorry
-- {(a, b) | (a = 0 β§ b = 0) β¨ 1 β€ |a| β¨ (0 < |a| β§ |a| < 1 β§ letI rm := (1 - β(1 - a ^ 2)) / a; letI rp := (1 + β(1 - a ^ 2)) / a; (b < Real.log (1 + rm ^ 2) - a * rm β¨ b > Real.log (1 + rp ^ 2) - a * rp))}
/--
Determine all ordered pairs of real numbers $(a,b)$ such that the line $y = ax+b$ intersects the curve $y = \ln(1+x^2)$ in exactly one point.
-/
theorem putnam_2022_a1
: {(a, b) | β! x : β, a * x + b = Real.log (1 + x^2)} = putnam_2022_a1_solution :=
sorry
|
putnam_1985_a1
|
Require Import Ensembles Finite_sets Nat.
Definition putnam_1985_a1_solution := (10, 10, 0, 0).
Theorem putnam_1985_a1
: let (abc, d) := putnam_1985_a1_solution in let (ab, c) := abc in let (a, b) := ab in
cardinal ((Ensemble nat) * (Ensemble nat) * (Ensemble nat)) (fun A => let (A1A2, A3) := A in let (A1, A2) := A1A2 in
Union nat (Union nat A1 A2) A3 = fun n => 1 <= n <= 10 /\ Intersection nat (Intersection nat A1 A2) A3 = Empty_set nat)
(2 ^ a * 3 ^ b * 5 ^ c * 7 ^ d).
Proof. Admitted.
|
import Mathlib
open Set
abbrev putnam_1985_a1_solution : β Γ β Γ β Γ β := sorry
-- (10, 10, 0, 0)
/--
Determine, with proof, the number of ordered triples $(A_1, A_2, A_3)$ of sets which have the property that
\begin{enumerate}
\item[(i)] $A_1 \cup A_2 \cup A_3 = \{1,2,3,4,5,6,7,8,9,10\}$, and
\item[(ii)] $A_1 \cap A_2 \cap A_3 = \emptyset$.
\end{enumerate}
Express your answer in the form $2^a 3^b 5^c 7^d$, where $a,b,c,d$ are nonnegative integers.
-/
theorem putnam_1985_a1 :
let (a, b, c, d) := putnam_1985_a1_solution;
{(A1, A2, A3) : Set β€ Γ Set β€ Γ Set β€ | A1 βͺ A2 βͺ A3 = Icc 1 10 β§ A1 β© A2 β© A3 = β
}.ncard = 2 ^ a * 3 ^ b * 5 ^ c * 7 ^ d :=
sorry
|
putnam_2022_a6
|
Require Import Nat Reals Coquelicot.Hierarchy. From mathcomp Require Import div fintype seq ssralg ssrbool ssrnat ssrnum .
Definition putnam_2022_a6_solution := fun n : nat => n.
Theorem putnam_2022_a6
(N : nat)
(M : nat)
(n := mul N 2)
(i0 : 'I_n)
(sumIntervals : ('I_n -> R) -> nat -> R := fun s k => sum_n (fun i => (((s (nth i0 (enum 'I_n) (i+1))))^(2*k-1) - ((s (nth i0 (enum 'I_n) i)))^(2*k-1))) (n-1))
(valid : nat -> ('I_n -> R) -> Prop := fun m s => forall (k: nat), and (le 1 k) (le k m) -> sumIntervals s k = 1)
(hvalid : nat -> Prop := fun m => exists (s : 'I_n -> R), (forall (i : 'I_n), (s i < s (ordS i)) /\ s (nth i0 (enum 'I_n) 0) > -1 /\ s (nth i0 (enum 'I_n) (n-1)) < 1) -> valid m s)
(hM : hvalid M)
(hMub : forall m : nat, hvalid m -> le m M)
: M = putnam_2022_a6_solution n.
Proof. Admitted.
|
import Mathlib
open Set
-- Note: uses (β β β) instead of (Fin (2 * n) β β)
abbrev putnam_2022_a6_solution : β β β := sorry
-- (fun n : β => n)
/--
Let $n$ be a positive integer. Determine, in terms of $n$, the largest integer $m$ with the following property: There exist real numbers $x_1,\dots,x_{2n}$ with $-1<x_1<x_2<\cdots<x_{2n}<1$ such that the sum of the lengths of the $n$ intervals $[x_1^{2k-1},x_2^{2k-1}],[x_3^{2k-1},x_4^{2k-1}],\dots,[x_{2n-1}^{2k-1},x_{2n}^{2k-1}]$ is equal to $1$ for all integers $k$ with $1 \leq k \leq m$.
-/
theorem putnam_2022_a6
(n : β) (hn : 0 < n) :
IsGreatest
{m : β | β x : β β β,
StrictMono x β§ -1 < x 1 β§ x (2 * n) < 1 β§
β k β Icc 1 m, β i β Icc 1 n, ((x (2 * i) : β) ^ (2 * k - 1) - (x (2 * i - 1)) ^ (2 * k - 1)) = 1}
(putnam_2022_a6_solution n) :=
sorry
|
putnam_1986_a6
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Definition putnam_1986_a6_solution : (nat -> nat) -> nat -> R := fun b n => (\prod_(1 <= i < n.+1) (b i)%:R) / n`!%:R.
Theorem putnam_1986_a6
(n : nat)
(npos : gt n 0)
(a : nat -> R)
(b : nat -> nat)
(bpos : forall i : nat, lt i n /\ gt n 0 -> gt (b i) 0)
(binj : forall i j : nat, lt i n /\ lt j n /\ gt i 0 /\ gt j 0 -> (b i = b j -> i = j))
(f : {poly R})
(hf : forall x : R, (1 - x) ^ n * f.[x] = 1 + \sum_(1 <= i < n.+1) ((a i) * x ^ (b i)))
: f.[1] = putnam_1986_a6_solution b n.
Proof. Admitted.
|
import Mathlib
open Real Equiv
noncomputable abbrev putnam_1986_a6_solution : (β β β) β β β β := sorry
-- fun b n β¦ (β i : Finset.Icc 1 n, b i) / Nat.factorial n
/--
Let $a_1, a_2, \dots, a_n$ be real numbers, and let $b_1, b_2, \dots, b_n$ be distinct positive integers. Suppose that there is a polynomial $f(x)$ satisfying the identity
\[
(1-x)^n f(x) = 1 + \sum_{i=1}^n a_i x^{b_i}.
\]
Find a simple expression (not involving any sums) for $f(1)$ in terms of $b_1, b_2, \dots, b_n$ and $n$ (but independent of $a_1, a_2, \dots, a_n$).
-/
theorem putnam_1986_a6
(n : β)
(npos : n > 0)
(a : β β β)
(b : β β β)
(bpos : β i β Finset.Icc 1 n, b i > 0)
(binj : β i β Finset.Icc 1 n, β j β Finset.Icc 1 n, b i = b j β i = j)
(f : Polynomial β)
(hf : β x : β, (1 - x) ^ n * f.eval x = 1 + β i : Finset.Icc 1 n, (a i) * x ^ (b i))
: (f.eval 1 = putnam_1986_a6_solution b n) :=
sorry
|
putnam_2004_b1
|
Require Import Nat Reals QArith Coquelicot.Coquelicot.
Theorem putnam_2004_b1
(c : nat -> Z)
(n : nat)
(r : Q)
(Preq0 : sum_n (fun i => IZR (c i) * (Q2R r) ^ i) n = 0)
: forall i : nat, lt i n -> exists m : Z, IZR m = sum_n (fun j => IZR (c (sub n j)) * (Q2R r) ^ (i + 1 - j)) i.
Proof. Admitted.
|
import Mathlib
open Nat Topology Filter
/--
Let $P(x)=c_nx^n+c_{n-1}x^{n-1}+\cdots+c_0$ be a polynomial with integer coefficients. Suppose that $r$ is a rational number such that $P(r)=0$. Show that the $n$ numbers $c_nr,\,c_nr^2+c_{n-1}r,\,c_nr^3+c_{n-1}r^2+c_{n-2}r,\dots,\,c_nr^n+c_{n-1}r^{n-1}+\cdots+c_1r$ are integers.
-/
theorem putnam_2004_b1
(n : β)
(P : Polynomial β€)
(r : β)
(Pdeg : P.degree = n)
(Preq0 : Polynomial.aeval r P = 0)
: β i β Finset.range n, β m : β€, m = β j β Finset.range (i + 1), (P.coeff (n - j) * r ^ (i + 1 - j)) :=
sorry
|
putnam_2013_b2
|
Require Import Ensembles Finite_sets Reals Coquelicot.Coquelicot.
Definition putnam_2013_b2_solution : R := 3.
Theorem putnam_2013_b2
(E: Ensemble (R -> R) := fun f => forall (x : R), exists (a : nat -> R) (N : nat), f x = 1 + sum_n_m (fun n => a n * cos (2 * PI * INR n * x)) 1 N /\ f x >= 0 /\
forall (n: nat), n mod 3 = 0%nat -> a n = 0)
(m: R)
(hm : exists f: R -> R, E f /\ f 0 = m)
(hmub : forall f : R -> R, E f -> f 0 <= m)
: m = putnam_2013_b2_solution.
Proof. Admitted.
|
import Mathlib
open Function Set
abbrev putnam_2013_b2_solution : β := sorry
-- 3
/--
Let $C = \bigcup_{N=1}^\infty C_N$, where $C_N$ denotes the set of those `cosine polynomials' of the form
\[
f(x) = 1 + \sum_{n=1}^N a_n \cos(2 \pi n x)
\]
for which:
\begin{enumerate}
\item[(i)]
$f(x) \geq 0$ for all real $x$, and
\item[(ii)]
$a_n = 0$ whenever $n$ is a multiple of $3$.
\end{enumerate}
Determine the maximum value of $f(0)$ as $f$ ranges through $C$, and
prove that this maximum is attained.
-/
theorem putnam_2013_b2
(CN : β β Set (β β β))
(hCN : β N : β, CN N =
{f : β β β |
(β x : β, f x β₯ 0) β§
β a : List β, a.length = N + 1 β§ (β n : Fin (N + 1), 3 β£ (n : β) β a[n]! = 0) β§
β x : β, f x = 1 + β n β Finset.Icc 1 N, a[(n : β)]! * Real.cos (2*Real.pi*n*x)}) :
IsGreatest {f 0 | f β β N β Ici 1, CN N} putnam_2013_b2_solution :=
sorry
|
putnam_2012_a6
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals normedtype topology sequences measure lebesgue_measure lebesgue_integral.
From mathcomp Require Import classical_sets.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
Variable R : realType.
Definition mu := [the measure _ _ of @lebesgue_measure R].
Definition putnam_2012_a6_solution : Prop := True.
Theorem putnam_2012_a6
(p : ((R * R) -> R) -> Prop)
(hp : forall f : (R * R) -> R, p f <->
continuous f /\ forall x1 x2 y1 y2 : R, x2 > x1 -> y2 > y1 -> (x2 - x1) * (y2 - y1) = 1 ->
\int[mu]_(x in [set x | x1 <= x <= x2]) (fun x' => \int[mu]_(y in [set y | y1 <= y <= y2]) f (x', y)) x = 0)
: (forall f, forall x y, p f -> f (x, y) = 0) <-> putnam_2012_a6_solution.
Proof. Admitted.
|
import Mathlib
open Matrix Function
-- Note: this formalization differs from the original problem wording in only allowing axis-aligned rectangles. The problem is solvable given this weaker hypothesis.
abbrev putnam_2012_a6_solution : Prop := sorry
-- True
/--
Let $f(x,y)$ be a continuous, real-valued function on $\mathbb{R}^2$. Suppose that, for every rectangular region $R$ of area $1$, the double integral of $f(x,y)$ over $R$ equals $0$.
Must $f(x,y)$ be identically $0$?
-/
theorem putnam_2012_a6
(p : ((β Γ β) β β) β Prop)
(hp : β f, p f β
Continuous f β§
β x1 x2 y1 y2 : β, x2 > x1 β y2 > y1
β (x2 - x1) * (y2 - y1) = 1 β β« x in x1..x2, β« y in y1..y2, f (x, y) = 0)
: ((β f x y, p f β f (x, y) = 0) β putnam_2012_a6_solution) :=
sorry
|
putnam_1964_a1
|
From mathcomp Require Import all_algebra all_ssreflect fintype.
From mathcomp Require Import reals.
From mathcomp Require Import classical_sets cardinality.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
Local Open Scope card_scope.
Variable R : realType.
Theorem putnam_1964_a1
(A : set (R*R))
(hAcard : A #= [set: 'I_6])
(dists : set R := [set d : R | exists a b : R*R, a \in A /\ b \in A /\ a != b /\ d = @Num.sqrt R ((fst a - fst b) ^+ 2 + (snd a - snd b) ^+ 2)])
: supremum 1 dists / infimum 1 dists >= @Num.sqrt R 3.
Proof. Admitted.
|
import Mathlib
/--
Let $A_1, A_2, A_3, A_4, A_5, A_6$ be distinct points in the plane. Let $D$ be the longest distance between any pair, and let $d$ the shortest distance. Show that $\frac{D}{d} \geq \sqrt 3$.
-/
theorem putnam_1964_a1
(A : Finset (EuclideanSpace β (Fin 2)))
(hAcard : A.card = 6)
(dists : Set β)
(hdists : dists = {d : β | β a b : EuclideanSpace β (Fin 2), a β A β§ b β A β§ a β b β§ d = dist a b})
: (sSup dists / sInf dists β₯ Real.sqrt 3) :=
sorry
|
putnam_2021_a4
|
Require Import Reals Coquelicot.Coquelicot.
Definition putnam_2021_a4_solution := (sqrt 2 / 2) * PI * ln 2 / ln 10.
Theorem putnam_2021_a4
(I : nat -> R := fun r => RInt (fun x => RInt (fun y => (1 + 2 * x ^ 2) / (1 + x ^ 4 + 6 * x ^ 2 * y ^ 2 + y ^ 4) - (1 + y ^ 2) / (2 + x ^ 4 + y ^ 4)) 0 (sqrt (INR r ^ 2 - x ^ 2))) 0 r)
: Lim_seq I = putnam_2021_a4_solution.
Proof. Admitted.
|
import Mathlib
open Filter Topology Metric
noncomputable abbrev putnam_2021_a4_solution : β := sorry
-- ((Real.sqrt 2) / 2) * Real.pi * Real.log 2
/--
Let
\[
I(R) = \iint_{x^2+y^2 \leq R^2} \left( \frac{1+2x^2}{1+x^4+6x^2y^2+y^4} - \frac{1+y^2}{2+x^4+y^4} \right)\,dx\,dy.
\]
Find
\[
\lim_{R \to \infty} I(R),
\]
or show that this limit does not exist.
-/
theorem putnam_2021_a4
(S : β β Set (EuclideanSpace β (Fin 2)))
(hS : S = fun R => ball (0 : EuclideanSpace β (Fin 2)) R)
(I : β β β)
(hI : I = fun R => β« p in S R,
(1 + 2*(p 0)^2)/(1 + (p 0)^4 + 6*(p 0)^2*(p 1)^2 + (p 1)^4) - (1 + (p 1)^2)/(2 + (p 0)^4 + (p 1)^4)) :
Tendsto I atTop (π putnam_2021_a4_solution) :=
sorry
|
putnam_1986_b4
|
Require Import Reals Ranalysis Coquelicot.Coquelicot.
Definition putnam_1986_b4_solution := True.
Theorem putnam_1986_b4
(G : R -> R)
(hGeq : forall (r: R), exists (m n: Z), G r = Rabs (r - sqrt (IZR (m ^ 2 + 2 * n ^ 2))))
(hGlb : forall (r: R), forall (m n: Z), G r <= Rabs (r - sqrt (IZR (m ^ 2 + 2 * n ^ 2))))
: filterlim G (Rbar_locally p_infty) (locally 0) <-> putnam_1986_b4_solution.
Proof. Admitted.
|
import Mathlib
open Real Equiv Polynomial Filter Topology
abbrev putnam_1986_b4_solution : Prop := sorry
-- True
/--
For a positive real number $r$, let $G(r)$ be the minimum value of $|r - \sqrt{m^2+2n^2}|$ for all integers $m$ and $n$. Prove or disprove the assertion that $\lim_{r\to \infty}G(r)$ exists and equals $0$.
-/
theorem putnam_1986_b4
(G : β β β)
(hGeq : β r : β, β m n : β€, G r = |r - sqrt (m ^ 2 + 2 * n ^ 2)|)
(hGlb : β r : β, β m n : β€, G r β€ |r - sqrt (m ^ 2 + 2 * n ^ 2)|)
: (Tendsto G atTop (π 0) β putnam_1986_b4_solution) :=
sorry
|
putnam_1971_a6
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals exp sequences topology.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Theorem putnam_1971_a6
(c : R)
(hc : forall n : int, n > 0 -> exists m : int, expR (c * ln n%:~R) = m%:~R)
: exists m : int, m >= 0 /\ c = m%:~R.
Proof. Admitted.
|
import Mathlib
open Set MvPolynomial
/--
Let $c$ be a real number such that $n^c$ is an integer for every positive integer $n$. Show that $c$ is a non-negative integer.
-/
theorem putnam_1971_a6
(c : β)
(hc : β n : β€, n > 0 β β m : β€, (n : β)^c = m)
: β m : β€, m β₯ 0 β§ c = m :=
sorry
|
putnam_2002_a1
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals normedtype sequences topology derive.
From mathcomp Require Import classical_sets.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
Variable R : realType.
Definition putnam_2002_a1_solution : nat -> nat -> R := fun k n : nat => (-k%:R) ^ n * n`!%:R.
Theorem putnam_2002_a1
(k : nat)
(P : nat -> {poly R})
(kpos : gt k 0)
(Pderiv : forall n : nat, forall x : R, derive1n n (fun x' : R => 1/(x' ^ k - 1)) x = (P n).[x] / ((x ^ k - 1) ^ (n.+1)))
: forall n : nat, (P n).[1] = putnam_2002_a1_solution k n.
Proof. Admitted.
|
import Mathlib
open Nat
abbrev putnam_2002_a1_solution : β β β β β := sorry
-- (fun k n : β => (-k) ^ n * (n)!)
/--
Let $k$ be a fixed positive integer. The $n$-th derivative of $\frac{1}{x^k-1}$ has the form $\frac{P_n(x)}{(x^k-1)^{n+1}}$ where $P_n(x)$ is a polynomial. Find $P_n(1)$.
-/
theorem putnam_2002_a1
(k : β)
(P : β β Polynomial β)
(kpos : k > 0)
(Pderiv : β n x, iteratedDeriv n (fun x' : β => 1 / (x' ^ k - 1)) x = ((P n).eval x) / ((x ^ k - 1) ^ (n + 1)))
: β n, (P n).eval 1 = putnam_2002_a1_solution k n :=
sorry
|
putnam_1969_a4
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals exp sequences topology normedtype measure lebesgue_measure lebesgue_integral.
From mathcomp Require Import classical_sets.
Import numFieldNormedType.Exports.
Import Order.TTheory GRing.Theory Num.Theory.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
Variable R : realType.
Definition mu := [the measure _ _ of @lebesgue_measure R].
Theorem putnam_1969_a4
(f : nat -> R := fun n => \sum_(1 <= i < n.+1) ((-1)^(i.+1) / ((i%:R) ^+ i)))
: f @ \oo --> (\int[mu]_(x in [set t : R | 0 < t < 1]) (expR (x * ln x))).
Proof. Admitted.
|
import Mathlib
open Matrix Filter Topology Set Nat
/--
Show that $\int_0^1 x^x dx = \sum_{n=1}^{\infty} (-1)^{n+1}n^{-n}$.
-/
theorem putnam_1969_a4
: Tendsto (fun n => β i β Finset.Icc (1 : β€) n, (-1)^(i+1)*(i : β)^(-i)) atTop (π (β« x in Ioo (0 : β) 1, x^x)) :=
sorry
|
putnam_1964_b2
|
From mathcomp Require Import all_algebra all_ssreflect fintype.
From mathcomp Require Import classical_sets cardinality.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope classical_set_scope.
Local Open Scope card_scope.
Theorem putnam_1964_b2
(A : finType)
(n : nat)
(npos : gt n 0)
(hn : [set : A] #= [set : 'I_n])
(P : set (set A))
(hPP : forall T U : set A, (T \in P /\ U \in P) -> T `&` U !=set0)
(hPS : ~exists T : set A, (T \notin P) /\ (forall U : set A, U \in P -> T `&` U !=set0))
: P #= [set : 'I_(2^(n - 1))].
Proof. Admitted.
|
import Mathlib
open Set Function Filter Topology
/--
Let $S$ be a finite set. A set $P$ of subsets of $S$ has the property that any two members of $P$ have at least one element in common and that $P$ cannot be extended (whilst keeping this property). Prove that $P$ contains exactly half of the subsets of $S$.
-/
theorem putnam_1964_b2
(S : Type*) [Fintype S] [Nonempty S]
(P : Finset (Set S))
(hPP : β T β P, β U β P, T β© U β β
)
(hPS : Β¬β T : Set S, T β P β§ (β U β P, T β© U β β
))
: (P.card = 2 ^ (Fintype.card S - 1)) :=
sorry
|
putnam_2010_b1
|
Require Import Reals Coquelicot.Coquelicot.
Definition putnam_2010_b1_solution := False.
Theorem putnam_2010_b1
: (exists (a: nat -> R), forall (m: nat), gt m 0 -> Series (fun i => (a i) ^ m) = (INR m)) <-> putnam_2010_b1_solution.
Proof. Admitted.
|
import Mathlib
open Filter Topology Set
abbrev putnam_2010_b1_solution : Prop := sorry
-- False
/--
Is there an infinite sequence of real numbers $a_1, a_2, a_3, \dots$ such that \[ a_1^m + a_2^m + a_3^m + \cdots = m \] for every positive integer $m$?
-/
theorem putnam_2010_b1
: (β a : β β β, β m : β, m > 0 β β' i : β, (a i)^m = m) β putnam_2010_b1_solution :=
sorry
|
putnam_2009_b5
|
Require Import Reals Coquelicot.Coquelicot.
Theorem putnam_2009_b5
(f: R -> R)
(hf : forall (x: R), 1 < x -> (ex_derive f x /\ Derive f x = (x ^ 2 - (f x) ^ 2) / (x ^ 2 * ((f x) ^ 2 + 1))))
: is_lim f p_infty p_infty.
Proof. Admitted.
|
import Mathlib
open Topology MvPolynomial Filter Set Metric
/--
Let $f: (1, \infty) \to \mathbb{R}$ be a differentiable function such that
\[
f'(x) = \frac{x^2 - f(x)^2}{x^2 (f(x)^2 + 1)}
\qquad \mbox{for all $x>1$.}
\]
Prove that $\lim_{x \to \infty} f(x) = \infty$.
-/
theorem putnam_2009_b5
(f : β β β)
(hfdiff : DifferentiableOn β f (Ioi 1))
(hf : β x > 1, deriv f x = (x ^ 2 - (f x) ^ 2) / ((x ^ 2) * ((f x) ^ 2 + 1)))
: (Tendsto f atTop atTop) :=
sorry
|
putnam_1971_a3
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals normedtype.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Theorem putnam_1971_a3
(a b c : R * R)
(r : R)
(habclattice : fst a = (floor (fst a))%:~R /\ snd a = (floor (snd a))%:~R /\ fst b = (floor (fst b))%:~R /\ snd b = (floor (snd b))%:~R /\ fst c = (floor (fst c))%:~R /\ snd c = (floor (snd c))%:~R)
(habcneq : a <> b /\ b <> c /\ c <> a)
(hr : r > 0)
(oncircle : (R * R) -> R -> (R * R) -> Prop := fun C rad p => (fst C - fst p)^+2 + (snd C - snd p)^+2 = rad^+2)
(hcircle : exists C : R * R, oncircle C r a /\ oncircle C r b /\ oncircle C r c)
: ((fst a - fst b)^+2 + (snd a - snd b)^+2) * ((fst b - fst c)^+2 + (snd b - snd c)^+2) * ((fst c - fst a)^+2 + (snd c - snd a)^+2) >= 4 * r ^+ 2.
Proof. Admitted.
|
import Mathlib
open Set
/--
The three vertices of a triangle of sides $a,b,c$ are lattice points and lie on a circle of radius $R$. Show that $abc \geq 2R$.
-/
theorem putnam_1971_a3
(a b c : β Γ β)
(R : β)
(habclattice : a.1 = round a.1 β§ a.2 = round a.2 β§ b.1 = round b.1 β§ b.2 = round b.2 β§ c.1 = round c.1 β§ c.2 = round c.2)
(habcneq : a β b β§ a β c β§ b β c)
(hR : R > 0)
(hcircle : β C : β Γ β,
β((a.1 - C.1)^2 + (a.2 - C.2)^2) = R β§
β((b.1 - C.1)^2 + (b.2 - C.2)^2) = R β§
β((c.1 - C.1)^2 + (c.2 - C.2)^2) = R) :
(β((a.1 - b.1)^2 + (a.2 - b.2)^2)) * (β((a.1 - c.1)^2 + (a.2 - c.2)^2)) * (β((b.1 - c.1)^2 + (b.2 - c.2)^2)) β₯ 2 * R :=
sorry
|
putnam_2009_a1
|
Require Import Reals GeoCoq.Main.Tarski_dev.Ch16_coordinates_with_functions.
Context `{T2D:Tarski_2D} `{TE:@Tarski_euclidean Tn TnEQD}.
Open Scope R.
Definition putnam_2009_a1_solution := True.
Theorem putnam_2009_a1
: (forall f: Tpoint -> R,
(forall (A B C D: Tpoint), Square A B C D ->
f A + f B + f C + f D = 0) ->
forall (P : Tpoint), f P = 0) <->
putnam_2009_a1_solution.
Proof. Admitted.
|
import Mathlib
open Topology MvPolynomial Filter
abbrev putnam_2009_a1_solution : Prop := sorry
-- True
/--
Let $f$ be a real-valued function on the plane such that for every square $ABCD$ in the plane, $f(A)+f(B)+f(C)+f(D)=0$. Does it follow that $f(P)=0$ for all points $P$ in the plane?
-/
theorem putnam_2009_a1
: ((β f : (β Γ β) β β, (β O v : β Γ β, v β (0, 0) β f (O.1, O.2) + f (O.1 + v.1, O.2 + v.2) + f (O.1 + v.1 - v.2, O.2 + v.2 + v.1) + f (O.1 - v.2, O.2 + v.1) = 0) β β P : β Γ β, f P = 0) β putnam_2009_a1_solution) :=
sorry
|
putnam_1971_a4
|
Require Import Nat Reals Coquelicot.Coquelicot.
Theorem putnam_1971_a4
(epsilon : R)
(hepsilon : 0 < epsilon < 1)
(P : nat -> (R * R) -> R := fun n '(x, y) => (x + y)^n * (x^2 - (2 - epsilon)*x*y + y^2))
: exists N : nat, forall n : nat, ge n N -> (exists (k : nat) (coeff : (nat * nat) -> R),
(forall i : nat * nat, coeff i >= 0) /\ (forall x : R * R, P n x = sum_n (fun i => sum_n (fun j => (coeff (i,j)) * (fst x)^i * (snd x)^j) k) k)).
Proof. Admitted.
(* Note: While I believe there is a mathcomp adjacent implementation for multinomials due to Strub, Hivert, and Thery, my understanding
is there are two representations there, and that there may be further work to unify them before integration into mathcomp. *)
|
import Mathlib
open Set MvPolynomial
/--
Show that for $\epsilon \in (0,1)$, the expression $(x + y)^n (x^2 - 2-\epsilon)xy + y^2)$ is a polynomial with positive coefficients for $n$ sufficiently large, where $n$ is an integer.
-/
theorem putnam_1971_a4
(Ξ΅ : β)
(hΞ΅ : 0 < Ξ΅ β§ Ξ΅ < 1)
(P : β β β β MvPolynomial (Fin 2) β)
(hP : P = fun (n : β) (Ξ΄ : β) => (MvPolynomial.X 0 + MvPolynomial.X 1)^n * ((MvPolynomial.X 0)^2 - (MvPolynomial.C (2 - Ξ΄))*(MvPolynomial.X 0)*(MvPolynomial.X 1) + (MvPolynomial.X 1)^2))
: β N : β, β n β₯ N, β i : Fin 2 ββ β, MvPolynomial.coeff i (P n Ξ΅) β₯ 0 :=
sorry
|
putnam_2010_b4
|
Require Import Reals Coquelicot.Coquelicot.
Definition P : (nat -> R) -> nat -> R -> R := fun c n x => sum_n (fun i => c i * x ^ i) n.
Definition putnam_2010_b4_solution (c1 c2: nat -> R) (n m: nat) := exists (a b c d: R), b * c - a * d = 1 /\ P c1 n = (fun x => a * x + b) /\ P c2 m = (fun x => c * x + d).
Theorem putnam_2010_b4
(c1 c2: nat -> R)
(n m: nat)
: (forall (x: R), (P c1 n x) * (P c2 m (x + 1)) - (P c1 n (x + 1)) * (P c2 m x) = 1) <-> putnam_2010_b4_solution c1 c2 n m.
Proof. Admitted.
|
import Mathlib
open Filter Topology Set
abbrev putnam_2010_b4_solution : Set (Polynomial β Γ Polynomial β) := sorry
-- {(p, q) : Polynomial β Γ Polynomial β | p.degree β€ 1 β§ q.degree β€ 1 β§ p.coeff 0 * q.coeff 1 - p.coeff 1 * q.coeff 0 = 1}
/--
Find all pairs of polynomials $p(x)$ and $q(x)$ with real coefficients for which $p(x)q(x+1)-p(x+1)q(x)=1$.
-/
theorem putnam_2010_b4
(p q : Polynomial β)
: (β x : β, p.eval x * q.eval (x + 1) - p.eval (x + 1) * q.eval x = 1) β (p, q) β putnam_2010_b4_solution :=
sorry
|
putnam_1973_b4
|
From mathcomp Require Import all_ssreflect ssrnum ssralg.
From mathcomp Require Import reals sequences topology measure lebesgue_measure lebesgue_integral normedtype derive.
From mathcomp Require Import classical_sets.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
Variable R : realType.
Definition mu := [the measure _ _ of @lebesgue_measure R].
Definition putnam_1973_b4_solution : R -> R := fun x => x.
Theorem putnam_1973_b4
(f : R -> R)
(hprop : (R -> R) -> Prop := fun g => (forall x : R, differentiable g x) /\ (continuous f^`()) /\ (forall x : R, 0 < f^`() x <= 1) /\ g 0 = 0)
(hf : hprop f)
: (\int[mu]_(x in [set x | 0 <= x <= 1]) f x) ^+ 2 >= \int[mu]_(x in [set x | 0 <= x <= 1]) ((f x) ^+ 3) /\
(\int[mu]_(x in [set x | 0 <= x <= 1]) putnam_1973_b4_solution x) ^+ 2 >= \int[mu]_(x in [set x | 0 <= x <= 1]) ((putnam_1973_b4_solution x) ^+ 3).
Proof. Admitted.
|
import Mathlib
open Nat Set MeasureTheory Topology Filter
-- Note: Boosted domain to β, which is fine because you can extend any such function f from [0,1] to β satisfying the same properties. There may be multiple correct answers.
abbrev putnam_1973_b4_solution : β β β := sorry
-- (fun x => x)
/--
Suppose $f$ is a function on $[0,1]$ with continuous derivative satisfying $0 < f'(x) \leq 1$ and $f 0 = 0$. Prove that $\left[\int_0^1 f(x) dx\right]]^2 \geq \int_0^1 (f(x))^3 dx$, and find an example where equality holds.
-/
theorem putnam_1973_b4
(f : β β β)
(hprop : (β β β) β Prop)
(hprop_def : hprop = fun g => ContDiff β 1 g β§ (β x : β, 0 < deriv g x β§ deriv g x β€ 1) β§ g 0 = 0)
(hf : hprop f)
: (β« x in Icc 0 1, f x)^2 β₯ β« x in Icc 0 1, (f x)^3 β§ (hprop putnam_1973_b4_solution β§ (β« x in Icc 0 1, putnam_1973_b4_solution x)^2 = β« x in Icc 0 1, (putnam_1973_b4_solution x)^3) :=
sorry
|
putnam_1981_a1
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals normedtype sequences topology.
From mathcomp Require Import classical_sets.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
Variable R : realType.
Definition putnam_1981_a1_solution : R := 1 / 8.
Theorem putnam_1981_a1
(P : nat -> nat -> Prop := fun n k => (5 ^ k %| (\prod_( 1<= i < n+1) (i%:Z ^+ i)))%Z)
(E : nat -> nat)
(hE : forall n : nat, ge n 1 -> P n (E n) /\ forall k : nat, P n k -> le k (E n))
: (fun n : nat => (E n)%:R / (n%:R ^ 2)) @ \oo --> putnam_1981_a1_solution.
Proof. Admitted.
|
import Mathlib
open Topology Filter Set Polynomial Function
noncomputable abbrev putnam_1981_a1_solution : β := sorry
-- 1/8
/--
Let $E(n)$ be the greatest integer $k$ such that $5^k$ divides $1^1 2^2 3^3 \cdots n^n$. Find $\lim_{n \rightarrow \infty} \frac{E(n)}{n^2}$.
-/
theorem putnam_1981_a1
(P : β β β β Prop)
(hP : β n k, P n k β 5^k β£ β m β Finset.Icc 1 n, (m^m : β€))
(E : β β β)
(hE : β n β Ici 1, P n (E n) β§ β k : β, P n k β k β€ E n) :
Tendsto (fun n : β => ((E n) : β)/n^2) atTop (π putnam_1981_a1_solution) :=
sorry
|
putnam_2005_b1
|
Require Import Reals Coquelicot.Coquelicot.
Definition putnam_2005_b1_solution := (fun x y => match x, y with | 0, 1 => -1 | 0, 2 => 1 | 1, 0 => 2 | 1, 1 => -4 | 2, 0 => 4 | _, _ => 0 end, (2%nat, 2%nat)).
Theorem putnam_2005_b1
(p : R -> R -> R := fun x y => sum_n (fun i => (sum_n (fun j => (fst putnam_2005_b1_solution) i j * x ^ i * y ^ j) (fst (snd putnam_2005_b1_solution)))) (snd (snd putnam_2005_b1_solution)))
: (~forall x y : R, p x y = 0) /\ forall (a : R), p (IZR (floor a)) (IZR (floor (2 * a))) = 0.
Proof. Admitted.
|
import Mathlib
open Nat Set
-- Note: There might be multiple possible correct answers.
noncomputable abbrev putnam_2005_b1_solution : MvPolynomial (Fin 2) β := sorry
-- (MvPolynomial.X 1 - 2 * MvPolynomial.X 0) * (MvPolynomial.X 1 - 2 * MvPolynomial.X 0 - 1)
/--
Find a nonzero polynomial $P(x,y)$ such that $P(\lfloor a \rfloor,\lfloor 2a \rfloor)=0$ for all real numbers $a$. (Note: $\lfloor \nu \rfloor$ is the greatest integer less than or equal to $\nu$.)
-/
theorem putnam_2005_b1
: putnam_2005_b1_solution β 0 β§ β a : β, MvPolynomial.eval (fun n : Fin 2 => if (n = 0) then (Int.floor a : β) else (Int.floor (2 * a))) putnam_2005_b1_solution = 0 :=
sorry
|
putnam_1980_b1
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals sequences.
From mathcomp Require Import classical_sets.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope classical_set_scope.
Local Open Scope ring_scope.
Variable R : realType.
Definition putnam_1980_b1_solution : set R := [set x : R | x >= 1/2].
Theorem putnam_1980_b1
(c : R)
: (forall x : R, (expR x + expR (-x)) / 2 <= expR (c * x^2)) <-> c \in putnam_1980_b1_solution.
Proof. Admitted.
|
import Mathlib
open Real
abbrev putnam_1980_b1_solution : Set β := sorry
-- {c : β | c β₯ 1 / 2}
/--
For which real numbers $c$ is $(e^x+e^{-x})/2 \leq e^{cx^2}$ for all real $x$?
-/
theorem putnam_1980_b1
(c : β)
: (β x : β, (exp x + exp (-x)) / 2 β€ exp (c * x ^ 2)) β c β putnam_1980_b1_solution :=
sorry
|
putnam_1980_b6
|
From mathcomp Require Import all_algebra all_ssreflect.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Theorem putnam_1980_b6
(G : (int * int)%type -> rat)
(hG : forall d n : nat, leq d n -> ((d = 1%nat -> G (d%:Z, n%:Z) = (1%:Q)/(n%:Q)) /\ (gtn d (1%nat) -> G (d%:Z, n%:Z) = ((d%:Z%:Q)/(n%:Z%:Q)) * \sum_(d <= i < n.+1) (G (d%:Z - 1, i%:Z - 1)))))
: forall (d p : nat), (ltn (1%nat) d /\ leq d p /\ prime p) -> ~ (p%:Z %| (denq (G (d%:Z, p%:Z))))%Z.
Proof. Admitted.
|
import Mathlib
open Set
/--
For integers $d, n$ with $1 \le d \le n$, let $G(1, n) = \frac{1}{n}$ and $G(d, n) = \frac{d}{n}\sum_{i=d}^{n}G(d - 1, i - 1)$ for all $d > 1$. If $1 < d \le p$ for some prime $p$, prove that the reduced denominator of $G(d, p)$ is not divisible by $p$.
-/
theorem putnam_1980_b6
(G : β€ Γ β€ β β)
(hG : β d n : β, d β€ n β (d = 1 β G (d, n) = 1/(n : β)) β§ (d > 1 β G (d, n) = (d/(n : β))*β i β Finset.Icc d n, G ((d : β€) - 1, (i : β€) - 1)))
: β d p : β, 1 < d β§ d β€ p β§ Prime p β Β¬p β£ (G (d, p)).den :=
sorry
|
putnam_1962_a2
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals normedtype sequences topology derive measure lebesgue_measure lebesgue_integral.
From mathcomp Require Import classical_sets.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
Variable R : realType.
Definition mu := [the measure _ _ of @lebesgue_measure R].
Definition putnam_1962_a2_solution : set (R -> R) := [set f | exists a c : R, a >= 0 /\ f = (fun x : R => a / (1 - c * x) ^ 2)].
Theorem putnam_1962_a2
(P : (set R) -> (R -> R) -> Prop)
(P_def : forall s f, P s f <-> ((forall x, f x >= 0) /\ forall x, x \in s ->
1/x * \int[mu]_(t in [set t | 0 <= t <= x]) f t = Num.sqrt (f 0 * f x)))
: (forall f,
(P [set t | 0 < t] f -> exists g, g \in putnam_1962_a2_solution /\ (forall x : R, x > 0 -> f x = g x)) /\
(forall e, 0 < e -> P [set t | 0 < t < e] f -> exists g, g \in putnam_1962_a2_solution /\ (forall x : R, 0 <= x < e -> f x = g x))) /\
forall f, f \in putnam_1962_a2_solution -> P [set t | 0 < t] f \/ exists e, 0 < e /\ P [set t | 0 < t < e] f.
Proof. Admitted.
|
import Mathlib
open MeasureTheory Set
abbrev putnam_1962_a2_solution : Set (β β β) := sorry
-- {f : β β β | β a c : β, a β₯ 0 β§ f = fun x β¦ a / (1 - c * x) ^ 2}
/--
Find every real-valued function $f$ whose domain is an interval $I$ (finite or infinite) having 0 as a left-hand endpoint, such that for every positive member $x$ of $I$ the average of $f$ over the closed interval $[0, x]$ is equal to the geometric mean of the numbers $f(0)$ and $f(x)$.
-/
theorem putnam_1962_a2
(P : Set β β (β β β) β Prop)
(P_def : β s f, P s f β 0 β€ f β§ β x β s, β¨ t in Ico 0 x, f t = β(f 0 * f x)) :
(β f,
(P (Ioi 0) f β β g β putnam_1962_a2_solution, EqOn f g (Ici 0)) β§
(β e > 0, P (Ioo 0 e) f β β g β putnam_1962_a2_solution, EqOn f g (Ico 0 e))) β§
β f β putnam_1962_a2_solution, P (Ioi 0) f β¨ (β e > 0, P (Ioo 0 e) f) :=
sorry
|
putnam_2015_a2
|
Require Import Nat Reals Znumtheory BinInt. From mathcomp Require Import div.
Definition putnam_2015_a2_solution : nat := 181.
Theorem putnam_2015_a2
(A := fix a (n: nat) : Z :=
match n with
| O => 1%Z
| S O => 2%Z
| S ((S n'') as n') => Z.sub (4*(a n')) (a n'')
end)
: odd putnam_2015_a2_solution = true /\ prime (Z.of_nat putnam_2015_a2_solution) /\ ((Z.of_nat putnam_2015_a2_solution) | A 2015%nat).
Proof. Admitted.
|
import Mathlib
-- Note: this problem admits several possible correct solutions; this is the one shown on the solutions document
abbrev putnam_2015_a2_solution : β := sorry
-- 181
/--
Let $a_0=1$, $a_1=2$, and $a_n=4a_{n-1}-a_{n-2}$ for $n \geq 2$. Find an odd prime factor of $a_{2015}$.
-/
theorem putnam_2015_a2
(a : β β β€)
(abase : a 0 = 1 β§ a 1 = 2)
(arec : β n β₯ 2, a n = 4 * a (n - 1) - a (n - 2))
: Odd putnam_2015_a2_solution β§ putnam_2015_a2_solution.Prime β§ ((putnam_2015_a2_solution : β€) β£ a 2015) :=
sorry
|
putnam_1990_a2
|
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1990_a2_solution := True.
Theorem putnam_1990_a2
(numform : R -> Prop := fun x => exists (n m: nat), x = Rpower (INR n) (1/3) - Rpower (INR m) (1/3))
: (exists (s: nat -> R), (forall (i: nat), numform (s i)) /\ is_lim_seq s (sqrt 2)) <-> putnam_1990_a2_solution.
Proof. Admitted.
|
import Mathlib
open Filter Topology Nat
abbrev putnam_1990_a2_solution : Prop := sorry
-- True
/--
Is $\sqrt{2}$ the limit of a sequence of numbers of the form $\sqrt[3]{n}-\sqrt[3]{m}$ ($n,m=0,1,2,\dots$)?
-/
theorem putnam_1990_a2
(numform : β β Prop)
(hnumform : β x : β, numform x β β n m : β, x = n ^ ((1 : β) / 3) - m ^ ((1 : β) / 3)) :
putnam_1990_a2_solution β
(β s : β β β,
(β i : β, numform (s i)) β§
Tendsto s atTop (π (Real.sqrt 2))) :=
sorry
|
putnam_2013_b5
|
Require Import Basics Reals Ensembles Finite_sets. From mathcomp Require Import fintype.
Theorem putnam_2013_b5
(n : nat)
(composen := fix compose_n (f : 'I_n -> 'I_n) (n : nat) :=
match n with
| O => fun x => x
| S n' => compose f (compose_n f n')
end)
(k : 'I_n)
(npos: ge n 1)
(E : Ensemble ('I_n -> 'I_n) := fun f => forall (x : 'I_n), exists (j : nat), (composen f j) x <= k)
: cardinal ('I_n -> 'I_n) E (k * n^(n-1)).
Proof. Admitted.
|
import Mathlib
open Function Set
/--
Let $X=\{1,2,\dots,n\}$, and let $k \in X$. Show that there are exactly $k \cdot n^{n-1}$ functions $f:X \to X$ such that for every $x \in X$ there is a $j \geq 0$ such that $f^{(j)}(x) \leq k$. [Here $f^{(j)}$ denotes the $j$\textsuperscript{th} iterate of $f$, so that $f^{(0)}(x)=x$ and $f^{(j+1)}(x)=f(f^{(j)}(x))$.]
-/
theorem putnam_2013_b5
(n : β) (hn : n β₯ 1)
(k : Set.Icc 1 n)
(fiter : (Set.Icc 1 n β Set.Icc 1 n) β Prop)
(hfiter : β f, fiter f β β x : Set.Icc 1 n, β j : β, f^[j] x β€ k) :
{f | fiter f}.encard = k * n ^ (n - 1) :=
sorry
|
putnam_2003_b1
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Definition putnam_2003_b1_solution : Prop := False.
Theorem putnam_2003_b1
: (exists a b c d : {poly R}, forall x y : R, 1 + x * y + x ^ 2 * y ^ 2 = a.[x] * c.[y] + b.[x] * d.[y]) <-> putnam_2003_b1_solution.
Proof. Admitted.
|
import Mathlib
open MvPolynomial Set
abbrev putnam_2003_b1_solution : Prop := sorry
-- False
/--
Do there exist polynomials $a(x), b(x), c(y), d(y)$ such that \[ 1 + xy + x^2y^2 = a(x)c(y) + b(x)d(y)\] holds identically?
-/
theorem putnam_2003_b1
: (β a b c d : Polynomial β, (β x y : β, 1 + x * y + x ^ 2 * y ^ 2 = a.eval x * c.eval y + b.eval x * d.eval y)) β putnam_2003_b1_solution :=
sorry
|
putnam_1974_a4
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals sequences exp.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Definition putnam_1974_a4_solution : nat -> R := fun n => (n%:R / 2 ^+ (n.-1)) * 'C(n.-1, n./2)%:R.
Theorem putnam_1974_a4
(n : nat)
(hn : gt n 0)
: 1/(2 ^ (n.-1)) * (\sum_(0 <= k < (n./2).+1) ((n%:R - 2 * k%:R) * 'C(n, k)%:R)) = putnam_1974_a4_solution n.
Proof. Admitted.
|
import Mathlib
open Set Nat
noncomputable abbrev putnam_1974_a4_solution : β β β := sorry
-- (fun n β¦ (1 : β) / (2 ^ (n - 1)) * (n * (n - 1).choose βn / 2ββ))
/--
Evaluate in closed form: $\frac{1}{2^{n-1}} \sum_{k < n/2} (n-2k)*{n \choose k}$.
-/
theorem putnam_1974_a4
(n : β)
(hn : 0 < n) :
(1 : β) / (2 ^ (n - 1)) * β k β Finset.Icc 0 βn / 2ββ, (n - 2 * k) * (n.choose k) =
putnam_1974_a4_solution n :=
sorry
|
putnam_1985_b3
|
Require Import Nat Ensembles Finite_sets.
Theorem putnam_1985_b3
(a : nat -> nat -> nat)
(apos : forall m n : nat, a m n > 0)
(ha : forall k : nat, k > 0 -> cardinal (nat * nat) (fun t => let (m, n) := t in m > 0 /\ n > 0 /\ a m n = k) 8)
: exists m n : nat, m > 0 /\ n > 0 /\ a m n > m * n.
Proof. Admitted.
|
import Mathlib
open Set Filter Topology Real Polynomial Function
/--
Let
\[
\begin{array}{cccc} a_{1,1} & a_{1,2} & a_{1,3} & \dots \\
a_{2,1} & a_{2,2} & a_{2,3} & \dots \\
a_{3,1} & a_{3,2} & a_{3,3} & \dots \\
\vdots & \vdots & \vdots & \ddots
\end{array}
\]
be a doubly infinite array of positive integers, and suppose each positive integer appears exactly eight times in the array. Prove that $a_{m,n} > mn$ for some pair of positive integers $(m,n)$.
-/
theorem putnam_1985_b3
(a : β β β β β)
(apos : β m n : β, a m n > 0)
(ha : β k : β, k > 0 β {(m, n) : β Γ β | m > 0 β§ n > 0 β§ a m n = k}.encard = 8)
: (β m n, m > 0 β§ n > 0 β§ a m n > m * n) :=
sorry
|
putnam_1996_a6
|
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1996_a6_solution (c: R) (f: R -> R) := if Rle_dec c (1/4) then (exists (d: R), f = (fun _ => d)) else ((forall (x: R), 0 <= x <= c -> continuity_pt f x) /\ f 0 = f c /\ (forall (x: R), x > 0 -> f x = f (pow x 2 + c)) /\ (forall (x: R), x < 0 -> f x = f (-x))).
Theorem putnam_1996_a6
(c: R)
(hc : c > 0)
: forall (f: R -> R), (continuity f /\ (forall (x: R), f x = pow x 2 + c)) <-> putnam_1996_a6_solution c f.
Proof. Admitted.
|
import Mathlib
open Function
abbrev putnam_1996_a6_solution : β β Set (β β β) := sorry
-- (fun c : β => if c β€ 1 / 4 then {f : β β β | β d : β, β x : β, f x = d} else {f : β β β | ContinuousOn f (Set.Icc 0 c) β§ f 0 = f c β§ (β x > 0, f x = f (x ^ 2 + c)) β§ (β x < 0, f x = f (-x))})
/--
Let $c>0$ be a constant. Give a complete description, with proof, of the set of all continuous functions $f:\mathbb{R} \to \mathbb{R}$ such that $f(x)=f(x^2+c)$ for all $x \in \mathbb{R}$.
-/
theorem putnam_1996_a6
(c : β)
(f : β β β)
(cgt0 : c > 0)
: (Continuous f β§ β x : β, f x = f (x ^ 2 + c)) β f β putnam_1996_a6_solution c :=
sorry
|
putnam_1971_b2
|
Require Import Reals Ensembles Coquelicot.Coquelicot.
Definition putnam_1971_b2_solution : Ensemble (R -> R) := fun f => (f = fun x : R => (x^3 - x^2 - 1)/(2 * x * (x-1))).
Theorem putnam_1971_b2
(S : Ensemble R := fun x => x <> 0 /\ x <> 1)
(P : (R -> R) -> Prop := fun F => forall x : R, In _ S x -> F x + F ((x-1)/x) = 1 + x)
: (forall F : R -> R, In _ putnam_1971_b2_solution F -> P F) /\ (forall f : R -> R, P f -> (exists F : R -> R, In _ putnam_1971_b2_solution F /\ (forall x : R, In _ S x -> f x = F x))).
Proof. Admitted.
|
import Mathlib
open Set MvPolynomial
abbrev putnam_1971_b2_solution : Set (β β β) := sorry
-- {fun x : β => (x^3 - x^2 - 1)/(2 * x * (x - 1))}
/--
Find all functions $F : \mathbb{R} \setminus \{0, 1\} \to \mathbb{R}$ that satisfy $F(x) + F\left(\frac{x - 1}{x}\right) = 1 + x$ for all $x \in \mathbb{R} \setminus \{0, 1\}$.
-/
theorem putnam_1971_b2
(S : Set β)
(hS : S = univ \ {0, 1})
(P : (β β β) β Prop)
(hP : P = fun (F : β β β) => β x β S, F x + F ((x - 1)/x) = 1 + x)
: (β F β putnam_1971_b2_solution, P F) β§ β f : β β β, P f β β F β putnam_1971_b2_solution, (β x β S, f x = F x) :=
sorry
|
putnam_1996_a5
|
Require Import Binomial Reals Znumtheory Coquelicot.Coquelicot. From mathcomp Require Import div.
Open Scope R.
Theorem putnam_1996_a5
(p : nat)
(hp : prime (Z.of_nat p) /\ gt p 3)
(k : nat := Z.to_nat (floor (2 * INR p / 3)))
: exists (m : nat), sum_n_m (fun i => Binomial.C p i) 1 k = INR m * pow (INR p) 2.
Proof. Admitted.
|
import Mathlib
open Function
/--
If $p$ is a prime number greater than 3 and $k = \lfloor 2p/3 \rfloor$, prove that the sum \[\binom p1 + \binom p2 + \cdots + \binom pk \] of binomial coefficients is divisible by $p^2$.
-/
theorem putnam_1996_a5
(p : β)
(hpprime : Prime p)
(hpge3 : p > 3)
(k : β)
(hk : k = Nat.floor (2*p/(3 : β)))
: p^2 β£ β i β Finset.Icc 1 k, Nat.choose p i :=
sorry
|
putnam_1963_b6
|
Require Import Ensembles. From GeoCoq Require Import Main.Tarski_dev.Ch16_coordinates_with_functions.
(* Note: This formalization assumes a 3D space; 1D and 2D spaces can be seen as lines and planes in this larger space. *)
Context `{T3D:Tarski_3D}.
Theorem putnam_1963_b6
(T : Ensemble Tpoint -> Ensemble Tpoint := fun A => (fun r => exists p q, In Tpoint A p /\ In Tpoint A q /\ Bet p r q))
(A : nat -> Ensemble Tpoint)
(hA0 : (A 0) <> Empty_set Tpoint)
(hAn : forall n : nat, n >= 1 -> A n = T (A (n - 1)))
: (forall n : nat, n >= 2 -> A n = A (n + 1)).
Proof. Admitted.
|
import Mathlib
open Topology Filter Polynomial
/--
Let $E$ be a Euclidean space of at most three dimensions. If $A$ is a nonempty subset of $E$, define $S(A)$ to be the set of all points that lie on closed segments joining pairs of points of $A$. For a given nonempty set $A_0$, define $A_n=S(A_{n-1})$ for $n=1,2,\dots$. Prove that $A_2=A_3=\cdots$. (A one-point set should be considered to be a special case of a closed segment.)
-/
theorem putnam_1963_b6
(d : β)
(S : Set (Fin d β β) β Set (Fin d β β))
(hS : S = fun A : Set (Fin d β β) => β p β A, β q β A, segment β p q)
(A : β β Set (Fin d β β))
(ddim : 1 β€ d β§ d β€ 3)
(hA0 : Nonempty (A 0))
(hAn : β n β₯ 1, A n = S (A (n - 1)))
: β n β₯ 2, A n = A (n + 1) :=
sorry
|
putnam_1978_a5
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals trigo.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Theorem putnam_1978_a5
(A : seq R)
(hanemp : gt (size A) 0)
(ha : all (fun x => 0 < x < pi) A)
(mu : R := \sum_(a <- A) (a / (size A)%:R))
: \prod_(a <- A) ((sin a)/a) <= (sin mu / mu) ^+ (size A).
Proof. Admitted.
|
import Mathlib
open Set Real
/--
Let $a_1, a_2, \dots , a_n$ be reals in the interval $(0, \pi)$ with arithmetic mean $\mu$. Show that
\[
\prod_{i=1}^n \left( \frac{\sin a_i}{a_i} \right) \leq \left( \frac{\sin \mu}{\mu} \right)^n.
\]
-/
theorem putnam_1978_a5
(n : β)
(npos : n > 0)
(a : Fin n β β)
(ha : β i : Fin n, a i β Ioo 0 Real.pi)
(ΞΌ : β)
(hΞΌ : ΞΌ = β i : Fin n, a i / n)
: (β i : Fin n, sin (a i) / (a i) β€ (sin ΞΌ / ΞΌ) ^ n) :=
sorry
|
putnam_1972_b3
|
From mathcomp Require Import fingroup.
Open Scope group_scope.
(* Note: This formalization is only for finite groups (due to mathcomp), but this is sufficiently general since the group generated by A and B is finite. *)
Variable T : finGroupType.
Theorem putnam_1972_b3
(G : {group T})
(A B : T)
(hab : A * B * A = B * A ^+ 2 * B /\ A ^+ 3 = 1 /\ exists n : nat, n > 0 /\ B ^+ (2*n - 1) = 1)
: (B = 1).
Proof. Admitted.
|
import Mathlib
open EuclideanGeometry Filter Topology Set MeasureTheory Metric
/--
Let $A$ and $B$ be two elements in a group such that $ABA = BA^2B$, $A^3 = 1$, and $B^{2n-1} = 1$ for some positive integer $n$. Prove that $B = 1$.
-/
theorem putnam_1972_b3
(G : Type*) [Group G]
(A B : G)
(hab : A * B * A = B * A^2 * B β§ A^3 = 1 β§ (β n : β€, n > 0 β§ B^(2*n - 1) = 1))
: B = 1 :=
sorry
|
putnam_1964_b3
|
From mathcomp Require Import all_ssreflect all_algebra.
From mathcomp Require Import reals sequences normedtype topology.
From mathcomp Require Import classical_sets.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
(* Note: There does not appear to be a clean way to express the eventually filter for types other than nats. *)
Variable R : realType.
Theorem putnam_1964_b3
(f : R -> R)
(hf : continuous f /\ forall alpha : R, alpha > 0 ->
(fun n : nat => f (n%:R * alpha)) @ \oo --> 0)
: f @ (filter_from setT (fun N => [set n | (N <= n)%R])) --> 0.
Proof. Admitted.
|
import Mathlib
open Set Function Filter Topology
/--
Suppose $f : \mathbb{R} \to \mathbb{R}$ is continuous and for every $\alpha > 0$, $\lim_{n \to \infty} f(n\alpha) = 0$. Prove that $\lim_{x \to \infty} f(x) = 0$.
-/
theorem putnam_1964_b3
(f : β β β)
(hf : Continuous f β§ β Ξ± > 0, Tendsto (fun n : β β¦ f (n * Ξ±)) atTop (π 0))
: (Tendsto f atTop (π 0)) :=
sorry
|
putnam_2000_a2
|
Require Import Reals.
Open Scope Z.
Theorem putnam_2000_a2
: forall (m: Z), exists (n: Z), n >= m /\
exists (a1 a2 b1 b2 c1 c2: Z), n = a1*a1 + a2*a2 /\ n+1 = b1*b1 + b2*b2 /\ n+2 = c1*c1 + c2*c2.
Proof. Admitted.
|
import Mathlib
open Topology Filter
/--
Prove that there exist infinitely many integers $n$ such that $n,n+1,n+2$ are each the sum of the squares of two integers.
-/
theorem putnam_2000_a2 :
β n : β,
β N : β€,
β i : Fin 6 β β,
N > n β§
N = (i 0)^2 + (i 1)^2 β§
N + 1 = (i 2)^2 + (i 3)^2 β§
N + 2 = (i 4)^2 + (i 5)^2 :=
sorry
|
putnam_2001_b3
|
Require Import Reals Coquelicot.Coquelicot.
Definition putnam_2001_b3_solution : R := 3.
Theorem putnam_2001_b3
(closest : nat -> R := (fun n : nat => IZR (floor (sqrt (INR n) + 0.5))))
: Series (fun n : nat => sum_n_m (fun n' : nat => (Rpower 2 (closest n') + Rpower 2 (-closest n')) / (2 ^ n')) 1 n) = putnam_2001_b3_solution.
Proof. Admitted.
|
import Mathlib
open Topology Filter Polynomial Set
abbrev putnam_2001_b3_solution : β := sorry
-- 3
/--
For any positive integer $n$, let $\langle n \rangle$ denote the closest integer to $\sqrt{n}$. Evaluate $\sum_{n=1}^\infty \frac{2^{\langle n \rangle}+2^{-\langle n \rangle}}{2^n}$.
-/
theorem putnam_2001_b3
: β' n : Set.Ici 1, ((2 : β) ^ (round (Real.sqrt n)) + (2 : β) ^ (-round (Real.sqrt n))) / 2 ^ (n : β) = putnam_2001_b3_solution :=
sorry
|
putnam_1985_a6
|
From mathcomp Require Import all_ssreflect all_algebra.
From mathcomp Require Import reals.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Open Scope ring_scope.
Variable R : realType.
Definition putnam_1985_a6_solution : {poly R} := 6 * 'X ^ 2 + 5 * 'X + 1.
Theorem putnam_1985_a6
(Gamma : {poly R} -> R := fun p => \sum_(i <- p) (i ^+ 2))
(f : {poly R} := 3 * 'X ^ 2 + 7 * 'X + 2)
: let g := putnam_1985_a6_solution in
g.[0] = 1 /\ forall n : nat, ge n 1 -> Gamma (f ^ n) = Gamma (g ^ n).
Proof. Admitted.
|
import Mathlib
open Set Filter Topology Real Polynomial
noncomputable abbrev putnam_1985_a6_solution : Polynomial β := sorry
-- 6 * X ^ 2 + 5 * X + 1
/--
If $p(x)= a_0 + a_1 x + \cdots + a_m x^m$ is a polynomial with real coefficients $a_i$, then set
\[
\Gamma(p(x)) = a_0^2 + a_1^2 + \cdots + a_m^2.
\]
Let $F(x) = 3x^2+7x+2$. Find, with proof, a polynomial $g(x)$ with real coefficients such that
\begin{enumerate}
\item[(i)] $g(0)=1$, and
\item[(ii)] $\Gamma(f(x)^n) = \Gamma(g(x)^n)$
\end{enumerate}
for every integer $n \geq 1$.
-/
theorem putnam_1985_a6
(Ξ : Polynomial β β β)
(f : Polynomial β)
(hΞ : Ξ = fun p β¦ β k β Finset.range (p.natDegree + 1), coeff p k ^ 2)
(hf : f = 3 * X ^ 2 + 7 * X + 2) :
let g := putnam_1985_a6_solution;
g.eval 0 = 1 β§ β n : β, n β₯ 1 β Ξ (f ^ n) = Ξ (g ^ n) :=
sorry
|
putnam_1966_a5
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals topology normedtype.
From mathcomp Require Import classical_sets.
Import numFieldTopology.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
Variable R : realType.
Theorem putnam_1966_a5
(C := [set f : R -> R | continuous f])
(T : (R -> R) -> (R -> R))
(imageTC : forall f : R -> R, f \in C -> T f \in C)
(linearT : forall a b : R, forall f g : R -> R, f \in C -> g \in C -> T (fun x : R => a * f x + b * g x) = (fun x => a * T f x + b * T g x))
(localT : forall r s : R, r <= s -> forall f g : R -> R, f \in C -> g \in C -> (forall x : R, r <= x <= s -> f x = g x) -> (forall x : R, r <= x <= s -> T f x = T g x))
: exists f : R -> R, f \in C /\ (forall g : R -> R, g \in C -> T g = fun x => f x * g x).
Proof. Admitted.
|
import Mathlib
open Topology Filter
/--
Let $C$ be the set of continuous functions $f : \mathbb{R} \to \mathbb{R}$. Let $T : C \to C$ satisfty the following two properties:
\begin{enumerate}
\item Linearity: $T(af + bg) = aT(f) + bT(g)$ for all $a, b \in \mathbb{R}$ and all $f, g \in C$.
\item Locality: If $f \in C$ and $g \in C$ are identical on some interval $I$, then $T(f)$ and $T(g)$ are also identical on $I$.
\end{enumerate}
Prove that there exists some function $f \in C$ such that $T(g(x)) = f(x)g(x)$ for all $g \in C$.
-/
theorem putnam_1966_a5
(C : Set (β β β))
(hC : C = {f : β β β | Continuous f})
(T : (β β β) β (β β β))
(imageTC : β f β C, T f β C)
(linearT : β a b : β, β f β C, β g β C, T ((fun x => a)*f + (fun x => b)*g) = (fun x => a)*(T f) + (fun x => b)*(T g))
(localT : β r s : β, r β€ s β β f β C, β g β C, (β x β Set.Icc r s, f x = g x) β (β x β Set.Icc r s, T f x = T g x))
: β f β C, β g β C, T g = f * g :=
sorry
|
putnam_2021_b2
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals sequences topology normedtype.
From mathcomp Require Import classical_sets.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
Variable R : realType.
Definition putnam_2021_b2_solution : R := 2/3.
Theorem putnam_2021_b2 :
putnam_2021_b2_solution \in supremums [set S : R | exists a : R ^nat, series a @ \oo --> 1 /\ (forall k, a k >= 0) /\
series (fun n : nat => n%:R / (2 ^+ n) * (\prod_(1 <= k < n.+1) a k) ^ (1 / n%:R)) @ \oo --> S].
Proof. Admitted.
|
import Mathlib
open Filter Topology
noncomputable abbrev putnam_2021_b2_solution : β := sorry
-- 2 / 3
/--
Determine the maximum value of the sum $S = \sum_{n=1}^\infty \frac{n}{2^n}(a_1a_2 \dots a_n)^{1/n}$ over all sequences $a_1,a_2,a_3,\dots$ of nonnegative real numbers satisfying $\sum_{k=1}^\infty a_k=1$.
-/
theorem putnam_2021_b2 :
IsGreatest
{S | β a : β+ β β, (β' k, a k = 1) β§ (β k, 0 β€ a k) β§
S = β' n : β+, n / 2 ^ (n : β) * (β k β Finset.Icc 1 n, a k) ^ (1 / n : β)}
putnam_2021_b2_solution :=
sorry
|
putnam_1988_b3
|
Require Import Reals Coquelicot.Coquelicot.
Definition putnam_1988_b3_solution := (1 + sqrt 3) / 2.
Theorem putnam_1988_b3
(r : Z -> R)
(hr : forall (n: Z), Z.ge n 1 -> (exists c d : Z, Z.ge c 0 /\ Z.ge d 0 /\ (Z.add c d) = n /\ r n = Rabs (IZR c - IZR d * sqrt 3)) /\ (forall c d : Z, (Z.ge c 0 /\ Z.ge d 0 /\ (Z.add c d) = n) -> Rabs (IZR c - IZR d * sqrt 3) >= r n))
: putnam_1988_b3_solution > 0 /\ (forall n : Z, Z.ge n 1 -> r n <= putnam_1988_b3_solution) /\ (forall (g: R), g > 0 -> (forall (n: Z), Z.ge n 1 -> r n <= g) -> g >= putnam_1988_b3_solution).
Proof. Admitted.
|
import Mathlib
open Set Filter Topology
noncomputable abbrev putnam_1988_b3_solution : β := sorry
-- (1 + Real.sqrt 3) / 2
/--
For every $n$ in the set $N=\{1,2,\dots\}$ of positive integers, let $r_n$ be the minimum value of $|c-d \sqrt{3}|$ for all nonnegative integers $c$ and $d$ with $c+d=n$. Find, with proof, the smallest positive real number $g$ with $r_n \leq g$ for all $n \in N$.
-/
theorem putnam_1988_b3
(r : β€ β β)
(hr : β n β₯ 1,
(β c d : β€,
(c β₯ 0 β§ d β₯ 0) β§
c + d = n β§ r n = |c - d * Real.sqrt 3|) β§
(β c d : β€, (c β₯ 0 β§ d β₯ 0 β§ c + d = n) β |c - d * Real.sqrt 3| β₯ r n))
: IsLeast {g : β | g > 0 β§ (β n : β€, n β₯ 1 β r n β€ g)} putnam_1988_b3_solution :=
sorry
|
putnam_1987_b2
|
Require Import Binomial Reals Coquelicot.Coquelicot.
Theorem putnam_1987_b2
: forall (n r s: nat), ge n (r + s) ->
sum_n (fun i => Binomial.C s i / Binomial.C n (r + i)) s = (INR n + 1)/((INR n + 1 - INR s) * Binomial.C (n - s) r).
Proof. Admitted.
|
import Mathlib
open MvPolynomial Real Nat
/--
Let $r, s$ and $t$ be integers with $0 \leq r$, $0 \leq s$ and $r+s \leq t$. Prove that
\[
\frac{\binom s0}{\binom tr}
+ \frac{\binom s1}{\binom{t}{r+1}} + \cdots
+ \frac{\binom ss}{\binom{t}{r+s}}
= \frac{t+1}{(t+1-s)\binom{t-s}{r}}.
\]
-/
theorem putnam_1987_b2
(r s t : β)
(hsum : r + s β€ t)
: (β i : Finset.range (s + 1), (choose s i : β) / (choose t (r + i)) = ((t + 1) : β) / ((t + 1 - s) * choose (t - s) r)) :=
sorry
|
putnam_2003_a2
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals exp sequences normedtype.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Theorem putnam_2003_a2
(n : nat)
(hn : gt n 0)
(a b : 'I_n -> R)
(abnneg : forall i : 'I_n, (a i) >= 0 /\ (b i) >= 0)
: expR (ln (\prod_(i < n) (a i)) * (1 / n%:R)) +
expR (ln (\prod_(i < n) (b i)) * (1 / n%:R)) <=
expR (ln (\prod_(i < n) (a i + b i)) * (1 / n%:R)).
Proof. Admitted.
|
import Mathlib
open MvPolynomial
/--
Let $a_1,a_2,\dots,a_n$ and $b_1,b_2,\dots,b_n$ be nonnegative real numbers. Show that $(a_1a_2 \cdots a_n)^{1/n}+(b_1b_2 \cdots b_n)^{1/n} \leq [(a_1+b_1)(a_2+b_2) \cdots (a_n+b_n)]^{1/n}$.
-/
theorem putnam_2003_a2
(n : β)
(hn : 0 < n)
(a b : Fin n β β)
(abnneg : β i, a i β₯ 0 β§ b i β₯ 0) :
(β i, a i) ^ ((1 : β) / n) +
(β i, b i) ^ ((1 : β) / n) β€
(β i, (a i + b i)) ^ ((1 : β) / n) :=
sorry
|
putnam_1980_a5
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals trigo lebesgue_integral lebesgue_measure measure.
From mathcomp Require Import classical_sets cardinality.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope classical_set_scope.
Local Open Scope ring_scope.
Variable R : realType.
Definition mu := [the measure _ _ of @lebesgue_measure R].
Theorem putnam_1980_a5
(P : {poly R})
(Pnonconst : gtn (size P) (1%nat))
: finite_set [set x : R | \int[mu]_(t in [set t : R | 0 <= t <= x]) (fun y => P.[y] * (sin y)) t = 0 /\ \int[mu]_(t in [set t : R | 0 <= t <= x]) (fun y => P.[y] * (cos y)) t = 0].
Proof. Admitted.
|
import Mathlib
/--
Let $P(t)$ be a nonconstant polynomial with real coefficients. Prove that the system of simultaneous equations $0=\int_0^xP(t)\sin t\,dt=\int_0^xP(t)\cos t\,dt$ has only finitely many real solutions $x$.
-/
theorem putnam_1980_a5
(P : Polynomial β)
(Pnonconst : P.degree > 0) :
Set.Finite {x : β |
0 = (β« t in (0)..x, P.eval t * Real.sin t) β§
0 = (β« t in (0)..x, P.eval t * Real.cos t)} :=
sorry
|
putnam_2003_b6
|
Require Import Reals Coquelicot.Coquelicot.
Theorem putnam_2003_b6
(f : R -> R)
(hf : continuity f)
: RInt (fun x => RInt (fun y => Rabs (f x + f y)) 0 1) 0 1 >= RInt (fun x => Rabs (f x)) 0 1.
Proof. Admitted.
|
import Mathlib
open MvPolynomial Set Nat
/--
Let $f(x)$ be a continuous real-valued function defined on the interval $[0,1]$. Show that \[ \int_0^1 \int_0^1 | f(x) + f(y) |\,dx\,dy \geq \int_0^1 |f(x)|\,dx. \]
-/
theorem putnam_2003_b6
(f : β β β)
(hf : Continuous f)
: (β« x in (0 : β)..1, (β« y in (0 : β)..1, |f x + f y|)) β₯ (β« x in (0 : β)..1, |f x|) :=
sorry
|
putnam_1985_a3
|
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1985_a3_solution (x: R) := exp x - 1.
Theorem putnam_1985_a3
(x: R)
(A := fix a (i j: nat) :=
match (i,j) with
| (i, 0) => x/pow 2 i
| (i, S j') => pow (a i j') 2 + 2 * a i j'
end)
: Lim_seq (fun n => A n n) = putnam_1985_a3_solution x.
Proof. Admitted.
|
import Mathlib
open Set Filter Topology Real
noncomputable abbrev putnam_1985_a3_solution : β β β := sorry
-- fun d β¦ exp d - 1
/--
Let $d$ be a real number. For each integer $m \geq 0$, define a sequence $\{a_m(j)\}$, $j=0,1,2,\dots$ by the condition
\begin{align*}
a_m(0) &= d/2^m, \\
a_m(j+1) &= (a_m(j))^2 + 2a_m(j), \qquad j \geq 0.
\end{align*}
Evaluate $\lim_{n \to \infty} a_n(n)$.
-/
theorem putnam_1985_a3
(d : β)
(a : β β β β β)
(ha0 : β m : β, a m 0 = d / 2 ^ m)
(ha : β m : β, β j : β, a m (j + 1) = (a m j) ^ 2 + 2 * a m j)
: Tendsto (fun n β¦ a n n) atTop (π (putnam_1985_a3_solution d)) :=
sorry
|
putnam_2021_a5
|
Require Import Nat. From mathcomp Require Import bigop div fintype eqtype seq ssrbool ssrnat.
Variables (I : finType) (P : pred I).
Definition putnam_2021_a5_solution (n: nat) := ~ (n %| 42 \/ n %| 46).
Theorem putnam_2021_a5
(A : pred 'I_2021 := fun n => let m := nat_of_ord n in ((1 <= m <= 2021) && (gcd m 2021 == 1)))
(B : nat -> nat := fun j => \sum_(n | A n) (nat_of_ord n)^j)
: forall (j: nat), B j mod 2021 = 0 <-> putnam_2021_a5_solution j.
Proof. Admitted.
|
import Mathlib
open Filter Topology
abbrev putnam_2021_a5_solution : Set β := sorry
-- {j : β | Β¬(42 β£ j) β§ Β¬(46 β£ j)}
/--
Let $A$ be the set of all integers $n$ such that $1 \leq n \leq 2021$ and $\gcd(n,2021)=1$. For every nonnegative integer $j$, let $S(j)=\sum_{n \in A}n^j$. Determine all values of $j$ such that $S(j)$ is a multiple of $2021$.
-/
theorem putnam_2021_a5
(j : β)
(A : Finset β)
(S : β β β)
(hA : A = {n | 1 β€ n β§ n β€ 2021 β§ Nat.gcd n 2021 = 1})
(hS : β j' : β, S j' = β n β A, n ^ j') :
(2021 β£ S j) β j β putnam_2021_a5_solution :=
sorry
|
putnam_2019_b2
|
Require Import Reals Coquelicot.Coquelicot.
Definition putnam_2019_b2_solution := 8 / PI ^ 3.
Theorem putnam_2019_b2
(a : nat -> R := fun n => sum_n_m (fun k => let k := INR k in let n := INR n in (sin (2 * (k + 1) * PI / (2 * n))) / ((cos ((k - 1) * PI / (2 * n))) ^ 2 * (cos ((k * PI) / (2 * n))) ^ 2)) 1 (n - 1))
: Lim_seq (fun n => a n / INR n ^ 3) = putnam_2019_b2_solution.
Proof. Admitted.
|
import Mathlib
open Topology Filter Set
noncomputable abbrev putnam_2019_b2_solution : β := sorry
-- 8/Real.pi^3
/--
For all $n \geq 1$, let
\[
a_n = \sum_{k=1}^{n-1} \frac{\sin \left( \frac{(2k-1)\pi}{2n} \right)}{\cos^2 \left( \frac{(k-1)\pi}{2n} \right) \cos^2 \left( \frac{k\pi}{2n} \right)}.
\]
Determine
\[
\lim_{n \to \infty} \frac{a_n}{n^3}.
\]
-/
theorem putnam_2019_b2
(a : β β β)
(ha : a = fun n : β => β k : Icc (1 : β€) (n - 1),
Real.sin ((2*k - 1)*Real.pi/(2*n))/((Real.cos ((k - 1)*Real.pi/(2*n))^2)*(Real.cos (k*Real.pi/(2*n))^2)))
: Tendsto (fun n : β => (a n)/n^3) atTop (π putnam_2019_b2_solution) :=
sorry
|
putnam_1993_b5
|
Require Import ZArith Reals Coquelicot.Coquelicot. From mathcomp Require Import fintype.
Theorem putnam_1993_b5
(pdists : ('I_4 -> (R * R)) -> Prop)
(hpdists : forall p : 'I_4 -> (R * R), pdists p = (forall i j : 'I_4, i <> j -> (exists k : Z, IZR k = norm (fst (p i) - fst (p j), (snd (p i) - snd (p j))) /\ Z.odd k = true)))
: ~ (exists p : 'I_4 -> (R * R), pdists p).
Proof. Admitted.
|
import Mathlib
/--
Show there do not exist four points in the Euclidean plane such that the pairwise distances between the points are all odd integers.
-/
theorem putnam_1993_b5:
Β¬β p : Fin 4 β (EuclideanSpace β (Fin 2)),
β i j, i β j β
(β n : β€, dist (p i) (p j) = n β§ Odd n) :=
sorry
|
putnam_1967_b2
|
From mathcomp Require Import all_ssreflect all_algebra.
From mathcomp Require Import reals normedtype.
From mathcomp Require Import classical_sets.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Theorem putnam_1967_b2
(p r A B C alpha beta gamma : R)
(prbound : 0 <= p <= 1 /\ 0 <= r <= 1)
(id1 : forall x y : R, (p * x + (1 - p) * y) ^+ 2 = A * x ^+ 2 + B * x * y + C * y ^+ 2)
(id2 : forall x y : R, (p * x + (1 - p) * y) * (r * x + (1 - r) * y) = alpha * x ^+ 2 + beta * x * y + gamma * y ^+ 2)
: sup [set A; B; C] >= 4/9 /\ sup [set alpha; beta; gamma] >= 4/9.
Proof. Admitted.
|
import Mathlib
open Nat Topology Filter
/--
Let $0 \leq p \leq 1$ and $0 \leq r \leq 1$ and consider the identities
\begin{enumerate}
\item[(a)] $(px+(1-p)y)^2=Ax^2+Bxy+Cy^2$,
\item[(b)] $(px+(1-p)y)(rx+(1-r)y)=\alpha x^2+\beta xy+\gamma y^2$.
\end{enumerate}
Show that (with respect to $p$ and $r$)
\begin{enumerate}
\item[(a)] $\max\{A,B,C\} \geq 4/9$,
\item[(b)] $\max\{\alpha,\beta,\gamma\} \geq 4/9$.
\end{enumerate}
-/
theorem putnam_1967_b2
(p r A B C Ξ± Ξ² Ξ³ : β)
(prbound : 0 β€ p β§ p β€ 1 β§ 0 β€ r β§ r β€ 1)
(id1 : β x y : β, (p * x + (1 - p) * y) ^ 2 = A * x ^ 2 + B * x * y + C * y ^ 2)
(id2 : β x y : β, (p * x + (1 - p) * y) * (r * x + (1 - r) * y) = Ξ± * x ^ 2 + Ξ² * x * y + Ξ³ * y ^ 2)
: max (max A B) C β₯ 4 / 9 β§ max (max Ξ± Ξ²) Ξ³ β₯ 4 / 9 :=
sorry
|
putnam_2023_b2
|
Require Import BinNums Nat NArith.
Definition putnam_2023_b2_solution := 3.
Theorem putnam_2023_b2
(k := fix count_ones (n : positive) : nat :=
match n with
| xH => 1
| xO n' => count_ones n'
| xI n' => 1 + count_ones n'
end)
: (forall (n: nat), n > 0 -> k (Pos.of_nat (2023*n)) >= putnam_2023_b2_solution) /\
(exists (n: nat), n > 0 /\ k (Pos.of_nat (2023*n)) = putnam_2023_b2_solution).
Proof. Admitted.
|
import Mathlib
open Nat
abbrev putnam_2023_b2_solution : β := sorry
-- 3
/--
For each positive integer $n$, let $k(n)$ be the number of ones in the binary representation of $2023 * n$. What is the minimum value of $k(n)$?
-/
theorem putnam_2023_b2
: sInf {(digits 2 (2023*n)).sum | n > 0} = putnam_2023_b2_solution :=
sorry
|
putnam_1968_b4
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals exp sequences topology normedtype measure lebesgue_measure lebesgue_integral.
From mathcomp Require Import classical_sets.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
Variable R : realType.
Definition mu := [the measure _ _ of @lebesgue_measure R].
Theorem putnam_1968_b4
(f : R -> R)
(hf : continuous f /\ (exists r : R, (fun y : nat => \int[mu]_(x in [set x : R | -(y%:R) <= x <= y%:R]) f x) @ \oo --> r))
: exists r : R,
(fun y : nat => \int[mu]_(x in [set x : R | -(y%:R) <= x <= y%:R]) f (x - 1 / x)) @ \oo --> r /\
(fun y : nat => \int[mu]_(x in [set x : R | -(y%:R) <= x <= y%:R]) f x) @ \oo --> r.
Proof. Admitted.
|
import Mathlib
open Finset Polynomial Topology Filter Metric
/--
Suppose that $f : \mathbb{R} \to \mathbb{R}$ is continuous on $(-\infty, \infty)$ and that $\int_{-\infty}^{\infty} f(x) dx$ exists. Prove that $$\int_{-\infty}^{\infty} f\left(x - \frac{1}{x}\right) dx = \int_{-\infty}^{\infty} f(x) dx.$$
-/
theorem putnam_1968_b4
(f : β β β)
(hf : Continuous f β§ β r : β, Tendsto (fun y => β« x in ball 0 y, f x) atTop (π r))
: β r : β, Tendsto (fun y => β« x in ball 0 y, f (x - 1/x)) atTop (π r) β§ Tendsto (fun y => β« x in ball 0 y, f x) atTop (π r) :=
sorry
|
putnam_2018_b2
|
Require Import Reals Coquelicot.Coquelicot. From Coqtail Require Import Cpow.
Open Scope C_scope.
Theorem putnam_2018_b2
(n : nat)
(hn : gt n 0)
(f : nat -> C -> C)
(hf : forall z : C, f n z = sum_n_m (fun i => (((RtoC (INR n)) - (RtoC (INR i))) * z ^ i)) 0 (n-1))
: forall (z : C), Cnorm z <= 1 -> f n z <> 0.
Proof. Admitted.
|
import Mathlib
/--
Let $n$ be a positive integer, and let $f_n(z) = n + (n-1)z + (n-2)z^2 + \cdots + z^{n-1}$. Prove that $f_n$ has no roots in the closed unit disk $\{z \in \mathbb{C}: |z| \leq 1\}$.
-/
theorem putnam_2018_b2
(n : β)
(hn : n > 0)
(f : β β β β β)
(hf : β z : β, f n z = β i β Finset.range n, (n - i) * z^i)
: β z : β, βzβ β€ 1 β f n z β 0 :=
sorry
|
putnam_2001_a5
|
Require Import Nat.
Theorem putnam_2001_a5
: exists! (a n: nat), a > 0 /\ n > 0 /\ a ^ (n + 1) - (a + 1) ^ n = 2001.
Proof. Admitted.
|
import Mathlib
open Topology Filter Polynomial Set
/--
Prove that there are unique positive integers $a$, $n$ such that $a^{n+1}-(a+1)^n=2001$.
-/
theorem putnam_2001_a5
: β! an : β€ Γ β, let (a, n) := an; a > 0 β§ n > 0 β§ a^(n+1) - (a+1)^n = 2001 :=
sorry
|
putnam_2016_b6
|
Require Import List Reals Coquelicot.Hierarchy Coquelicot.Series.
Definition putnam_2016_b6_solution := 1.
Theorem putnam_2016_b6:
Series (fun k => (-1)^k/(INR k+1) * Series (fun n => 1/(INR k*(2^n)+1))) = putnam_2016_b6_solution.
Proof. Admitted.
|
import Mathlib
open Polynomial Filter Topology Real Set Nat List
abbrev putnam_2016_b6_solution : β := sorry
-- 1
/--
Evaluate $\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k} \sum_{n=0}^\infty \frac{1}{k2^n+1}$.
-/
theorem putnam_2016_b6 :
β' k : β, ((-1 : β) ^ ((k + 1 : β€) - 1) / (k + 1 : β)) * β' n : β, (1 : β) / ((k + 1) * (2 ^ n) + 1) = putnam_2016_b6_solution :=
sorry
|
putnam_1974_a6
|
From mathcomp Require Import all_algebra all_ssreflect.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Definition putnam_1974_a6_solution : nat := 25%nat.
Theorem putnam_1974_a6
(hdivnallx : {poly int} -> Prop := fun f => (f \is monic) /\ (forall x : int, ((10^6) %| f.[x])%Z))
: (exists f : {poly int}, hdivnallx f /\ size f = putnam_1974_a6_solution.+1) /\
(forall d : nat, lt d putnam_1974_a6_solution -> ~ (exists f : {poly int}, hdivnallx f /\ size f = d.+1)).
Proof. Admitted.
|
import Mathlib
open Set Nat Polynomial
abbrev putnam_1974_a6_solution : β := sorry
-- 25
/--
Given $n$, let $k(n)$ be the minimal degree of any monic integral polynomial $f$ such that the value of $f(x)$ is divisible by $n$ for every integer $x$. Find the value of $k(1000000)$.
-/
theorem putnam_1974_a6
(hdivnallx : Polynomial β€ β Prop)
(hdivnallx_def : hdivnallx = fun f => Monic f β§ (β x : β€, (10^6 : β€) β£ f.eval x))
: sInf {d : β | β f : Polynomial β€, hdivnallx f β§ d = f.natDegree} = putnam_1974_a6_solution :=
sorry
|
putnam_1984_b1
|
Require Import Factorial ZArith.
Open Scope Z.
Fixpoint nat_sum (a : nat -> nat) (k : nat) : nat :=
match k with
| O => a O
| S k' => a k + nat_sum a k'
end.
Fixpoint Z_sum (a : nat -> Z) (k : nat) : Z :=
match k with
| O => a O
| S k' => a k + Z_sum a k'
end.
Definition putnam_1984_b1_solution : (nat -> Z) * (nat -> Z) := (fun x => match x with | O => 3 | S O => 1 | _ => 0 end, fun x => match x with | O => -2 | S O => -1 | _ => 0 end).
Theorem putnam_1984_b1
(f : nat -> Z)
(poly : nat -> (nat -> Z) -> nat -> Z := fun k p n => Z_sum (fun i => (p i) * (Z.of_nat (n ^ i))) k)
(deg : (nat -> Z) -> nat)
(finpoly : (nat -> Z) -> Prop := fun p => (exists k : nat, p k <> 0) /\ (exists m : nat, forall n : nat, gt n m -> p n = 0))
(hdeg : forall p : nat -> Z, finpoly p -> (p (deg p) <> 0 /\ forall n : nat, gt n (deg p) -> p n = 0))
(hf : forall n : nat, gt n 0 -> f n = (Z_sum (fun i => Z.of_nat (fact i)) n) - Z.of_nat (fact 0))
: finpoly (fst putnam_1984_b1_solution) /\ finpoly (snd putnam_1984_b1_solution) /\
forall n : nat, ge n 1 -> f (Nat.add n 2) = poly (deg (fst putnam_1984_b1_solution)) (fst putnam_1984_b1_solution) n * f (Nat.add n 1) + poly (deg (snd putnam_1984_b1_solution)) (snd putnam_1984_b1_solution) n * f n.
Proof. Admitted.
|
import Mathlib
open Topology Filter Nat
-- Note: This problem may have multiple correct answers.
noncomputable abbrev putnam_1984_b1_solution : Polynomial β Γ Polynomial β := sorry
-- (Polynomial.X + 3, -Polynomial.X - 2)
/--
Let $n$ be a positive integer, and define $f(n)=1!+2!+\dots+n!$. Find polynomials $P(x)$ and $Q(x)$ such that $f(n+2)=P(n)f(n+1)+Q(n)f(n)$ for all $n \geq 1$.
-/
theorem putnam_1984_b1
(f : β β β€)
(hf : β n > 0, f n = β i : Set.Icc 1 n, ((i)! : β€))
: let (P, Q) := putnam_1984_b1_solution; β n β₯ 1, f (n + 2) = P.eval (n : β) * f (n + 1) + Q.eval (n : β) * f n :=
sorry
|
putnam_2006_b2
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Open Scope ring_scope.
Variable R : realType.
Theorem putnam_2006_b2
(n : nat)
(hn : gt n 0)
(X : seq R)
(hX : uniq X /\ size X = n)
: exists S : seq R, subseq S X /\
size S <> 0%nat /\
(exists m : int, `|m%:~R + \sum_(s <- S) s| <= 1 / (n%:R + 1)).
Proof. Admitted.
|
import Mathlib
/--
Prove that, for every set $X = \{x_1, x_2, \dots, x_n\}$ of $n$ real numbers, there exists a non-empty subset $S$ of $X$ and an integer $m$ such that
\[
\left| m + \sum_{s \in S} s \right| \leq \frac{1}{n+1}.
\]
-/
theorem putnam_2006_b2
(n : β)
(npos : n > 0)
(X : Finset β)
(hXcard : X.card = n)
: (β S β X, S β β
β§ β m : β€, |m + β s β S, s| β€ 1 / (n + 1)) :=
sorry
|
putnam_1966_a1
|
From mathcomp Require Import all_ssreflect ssrnum ssralg ssrint.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Theorem putnam_1966_a1
(f : nat -> int := fun n => \sum_(0 <= m < n + 1) (if (~~odd m) then (m%:Z)/2 else (m%:Z-1)/2))
: forall x y : nat, (gt x 0) -> (gt y 0) -> gt x y -> (x * y)%:Z = f (Nat.add x y) - f (Nat.sub x y).
Proof. Admitted.
|
import Mathlib
/--
Let $a_n$ denote the sequence $0, 1, 1, 2, 2, 3, \dots$, where $a_n = \frac{n}{2}$ if $n$ is even and $\frac{n - 1}{2}$ if n is odd. Furthermore, let $f(n)$ denote the sum of the first $n$ terms of $a_n$. Prove that all positive integers $x$ and $y$ with $x > y$ satisfy $xy = f(x + y) - f(x - y)$.
-/
theorem putnam_1966_a1
(f : β€ β β€)
(hf : f = fun n : β€ => β m β Finset.Icc 0 n, (if Even m then m / 2 else (m - 1)/2))
: β x y : β€, x > 0 β§ y > 0 β§ x > y β x * y = f (x + y) - f (x - y) :=
sorry
|
putnam_2012_a3
|
Require Import Reals Coquelicot.Coquelicot.
Definition putnam_2012_a3_solution (x: R) := sqrt (1 - x ^ 2).
Theorem putnam_2012_a3
(hf : (R -> R) -> Prop := fun f : R -> R => continuous_on (fun x : R => -1 <= x <= 1) f /\ (forall x : R, -1 <= x <= 1 -> f x = ((2 - x^2)/2)*f (x^2/(2 - x^2))) /\ f 0 = 1 /\ exists (y : R), filterlim (fun x => f x / sqrt (1 - x)) (at_left 1) (locally y))
: hf putnam_2012_a3_solution /\ (forall f : R -> R, hf f -> (forall x : R, -1 <= x <= 1 -> f x = putnam_2012_a3_solution x)).
Proof. Admitted.
|
import Mathlib
open Matrix Function
-- Note: uses (β β β) instead of (Set.Icc (-1 : β) 1 β β)
noncomputable abbrev putnam_2012_a3_solution : β β β := sorry
-- fun x : β => Real.sqrt (1 - x^2)
/--
Let $f: [-1, 1] \to \mathbb{R}$ be a continuous function such that
\begin{itemize}
\item[(i)]
$f(x) = \frac{2-x^2}{2} f \left( \frac{x^2}{2-x^2} \right)$ for every $x$ in $[-1, 1]$,
\item[(ii)]
$f(0) = 1$, and
\item[(iii)]
$\lim_{x \to 1^-} \frac{f(x)}{\sqrt{1-x}}$ exists and is finite.
\end{itemize}
Prove that $f$ is unique, and express $f(x)$ in closed form.
-/
theorem putnam_2012_a3
(S : Set β)
(hS : S = Set.Icc (-1 : β) 1)
(fsat : (β β β) β Prop)
(hfsat : fsat = fun f : β β β => ContinuousOn f S β§
(β x β S, f x = ((2 - x^2)/2)*f (x^2/(2 - x^2))) β§ f 0 = 1 β§
(β y : β, leftLim (fun x : β => (f x)/Real.sqrt (1 - x)) 1 = y))
: fsat putnam_2012_a3_solution β§ β f : β β β, fsat f β β x β S, f x = putnam_2012_a3_solution x :=
sorry
|
putnam_1979_a6
|
From mathcomp Require Import all_algebra all_ssreflect fintype.
From mathcomp Require Import reals.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Theorem putnam_1979_b6
(p : seq R)
(hp : all (fun x => 0 <= x <= 1) p)
: exists x : R, 0 <= x <= 1 /\ (all (fun i => x != i) p) /\ (\sum_(i <- p) 1/`|x - i|) <= 8*(size p)%:R*(\sum_(0 <= i < (size p).+1) (1%R)/(2*(i%:R) + 1)).
Proof. Admitted.
|
import Mathlib
open Set
/--
For all $i \in \{0, 1, \dots, n - 1\}$, let $p_i \in [0, 1]$. Prove that there exists some $x \in [0, 1]$ such that $$\sum_{i = 0}^{n - 1} \frac{1}{|x - p_i|} \le 8n\left(\sum_{i = 0}^{n-1} \frac{1}{2i + 1}\right).$$
-/
theorem putnam_1979_a6
(n : β)
(p : β β β)
(hp : β i β Finset.range n, p i β Icc 0 1)
: β x β Icc 0 1, (β i β Finset.range n, x β p i) β§ β i β Finset.range n, 1/|x - p i| β€ 8*n*β i β Finset.range n, (1 : β)/(2*i + 1) :=
sorry
|
putnam_1978_a3
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals measure lebesgue_integral lebesgue_measure topology normedtype exp sequences.
From mathcomp Require Import classical_sets.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
Variable R : realType.
Definition mu := [the measure _ _ of @lebesgue_measure R].
Definition putnam_1978_a3_solution : nat := 2.
Theorem putnam_1978_a3
(p : {poly R} := 2 *: ('X^6 + 1%:P) + 4 *: ('X^5 + 'X) + 3 *: ('X^4 + 'X^2) + 5 *: 'X^3)
(I : nat -> R := fun k => \int[mu]_(x in [set t : R | 0 <= t]) (fun x => x ^ k / p.[x]) x)
: putnam_1978_a3_solution \in index_iota 1 5 /\ (forall k : nat, k \in index_iota 1 5 -> I putnam_1978_a3_solution <= I k).
Proof. Admitted.
|
import Mathlib
open Set Polynomial
abbrev putnam_1978_a3_solution : β := sorry
-- 2
/--
Let $p(x) = 2(x^6 + 1) + 4(x^5 + x) + 3(x^4 + x^2) + 5x^3$. For $k$ with $0 < k < 5$, let
\[
I_k = \int_0^{\infty} \frac{x^k}{p(x)} \, dx.
\]
For which $k$ is $I_k$ smallest?
-/
theorem putnam_1978_a3
(p : Polynomial β)
(hp : p = 2 * (X ^ 6 + 1) + 4 * (X ^ 5 + X) + 3 * (X ^ 4 + X ^ 2) + 5 * X ^ 3)
(I : β β β)
(hI : I = fun k β¦ β« x in Ioi 0, x ^ k / p.eval x) :
IsLeast {y | β k β Ioo 0 5, I k = y} putnam_1978_a3_solution :=
sorry
|
putnam_1987_b1
|
Require Import Reals Coquelicot.Coquelicot.
Definition putnam_1987_b1_solution := 1.
Theorem putnam_1987_b1
: RInt (fun x => ln (9 - x) ^ (1/2) / ( ln (9 - x) ^ (1/2) + ln (x + 3) ^ (1/2))) 2 4 = putnam_1987_b1_solution.
Proof. Admitted.
|
import Mathlib
open MvPolynomial Real Nat
abbrev putnam_1987_b1_solution : β := sorry
-- 1
/--
Evaluate
\[
\int_2^4 \frac{\sqrt{\ln(9-x)}\,dx}{\sqrt{\ln(9-x)}+\sqrt{\ln(x+3)}}.
\]
-/
theorem putnam_1987_b1
: (β« x in (2)..4, sqrt (log (9 - x)) / (sqrt (log (9 - x)) + sqrt (log (x + 3))) = putnam_1987_b1_solution) :=
sorry
|
putnam_1963_a2
|
From mathcomp Require Import all_algebra all_ssreflect.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope nat_scope.
Theorem putnam_1963_a2
(f : nat -> nat)
(hfpos : forall n : nat, 0 < f n)
(hfinc : forall i j : nat, 0 < i -> i < j -> f i < f j)
(hf2 : f 2 = 2)
(hfmn : forall m n : nat, 0 < m -> 0 < n -> coprime m n -> f (m * n) = f m * f n)
: forall n : nat, 0 < n -> f n = n.
Proof. Admitted.
|
import Mathlib
open Topology Filter
/--
Let $\{f(n)\}$ be a strictly increasing sequence of positive integers such that $f(2)=2$ and $f(mn)=f(m)f(n)$ for every relatively prime pair of positive integers $m$ and $n$ (the greatest common divisor of $m$ and $n$ is equal to $1$). Prove that $f(n)=n$ for every positive integer $n$.
-/
theorem putnam_1963_a2
(f : β β β)
(hfpos : β n, f n > 0)
(hfinc : StrictMonoOn f (Set.Ici 1))
(hf2 : f 2 = 2)
(hfmn : β m n, m > 0 β n > 0 β IsRelPrime m n β f (m * n) = f m * f n)
: β n > 0, f n = n :=
sorry
|
putnam_1962_b2
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals.
From mathcomp Require Import classical_sets.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Open Scope ring_scope.
Open Scope classical_set_scope.
Variable R : realType.
Theorem putnam_1962_b2
: exists f : R -> set nat, forall a b : R, a < b -> f a `<` f b.
Proof. Admitted.
|
import Mathlib
open MeasureTheory
--Note: The original problem requires a function to be exhibited, but in the official archives the solution depends on an enumeration of the rationals, so we modify the problem to be an existential statement.
/--
Let $\mathbb{S}$ be the set of all subsets of the natural numbers. Prove the existence of a function $f : \mathbb{R} \to \mathbb{S}$ such that $f(a) \subset f(b)$ whenever $a < b$.
-/
theorem putnam_1962_b2
: β f : β β Set β+, β a b : β, a < b β f a β f b :=
sorry
|
putnam_1975_a1
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals.
From mathcomp Require Import classical_sets.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
Variable R : realType.
Definition putnam_1975_a1_solution : ((int * int) -> int) * ((int * int) -> int) := (fun '(a, b) => a + b + 1, fun '(a, b) => a - b).
Theorem putnam_1975_a1
(nab : set (int * int * int) := [set nab | let '(n, a, b) := nab in n = (a%:~R ^+ 2 + a%:~R)/2 + (b%:~R ^+ 2 + b%:~R)/2])
(nxy : set (int * int * int) := [set nxy | let '(n, x, y) := nxy in 4 * n + 1 = x ^+ 2 + y ^+ 2])
: (forall n a b : int, nab (n, a, b) <-> nxy (n, putnam_1975_a1_solution.1 (a, b), putnam_1975_a1_solution.2 (a, b))) /\
forall n : int, (exists x y : int, nxy (n, x, y)) -> exists a b : int, nab (n, a, b).
Proof. Admitted.
|
import Mathlib
open Polynomial
abbrev putnam_1975_a1_solution : ((β€ Γ β€) β β€) Γ ((β€ Γ β€) β β€) := sorry
-- (fun (a, b) => a + b + 1, fun (a, b) => a - b)
/--
If an integer $n$ can be written as the sum of two triangular numbers (that is, $n = \frac{a^2 + a}{2} + \frac{b^2 + b}{2}$ for some integers $a$ and $b$), express $4n + 1$ as the sum of the squares of two integers $x$ and $y$, giving $x$ and $y$ in terms of $a$ and $b$. Also, show that if $4n + 1 = x^2 + y^2$ for some integers $x$ and $y$, then $n$ can be written as the sum of two triangular numbers.
-/
theorem putnam_1975_a1
(nab nxy : (β€ Γ β€ Γ β€) β Prop)
(hnab : nab = fun (n, a, b) => n = (a^2 + (a : β))/2 + (b^2 + (b : β))/2)
(hnxy : nxy = fun (n, x, y) => 4*n + 1 = x^2 + y^2)
: (β n a b : β€, nab (n, a, b) β nxy (n, putnam_1975_a1_solution.1 (a, b), putnam_1975_a1_solution.2 (a, b))) β§
β n : β€, (β x y : β€, nxy (n, x, y)) β β a b : β€, nab (n, a, b) :=
sorry
|
putnam_2016_b5
|
Require Import Reals Rpower.
Open Scope R.
Definition putnam_2016_b5_solution : (R -> R) -> Prop := fun f => exists (c: R), c > 0 /\ forall (x: R), x > 1 -> f x = Rpower x c .
Theorem putnam_2016_b5
(f : R -> R)
(fle : Prop := (forall x : R, x > 1 -> f x > 1) /\ (forall x y : R, (x > 1 /\ y > 1 /\ x*x <= y <= x*x*x) -> ((f x) * (f x) <= f y <= (f x) * (f x) * (f x))))
: fle <-> putnam_2016_b5_solution f.
Proof. Admitted.
|
import Mathlib
open Polynomial Filter Topology Real Set Nat List
abbrev putnam_2016_b5_solution : Set (Set.Ioi (1 : β) β Set.Ioi (1 : β)) := sorry
-- {f : Set.Ioi (1 : β) β Set.Ioi (1 : β) | β c : β, c > 0 β§ β x : Set.Ioi (1 : β), (f x : β) = x ^ c}
/--
Find all functions $f$ from the interval $(1,\infty)$ to $(1,\infty)$ with the following property: if $x,y \in (1,\infty)$ and $x^2 \leq y \leq x^3$, then $(f(x))^2 \leq f(y) \leq (f(x))^3$.
-/
theorem putnam_2016_b5
(f : Set.Ioi (1 : β) β Set.Ioi (1 : β)) :
f β putnam_2016_b5_solution β
(β x y : Set.Ioi (1 : β), ((x : β) ^ 2 β€ y β§ y β€ (x : β) ^ 3) β ((f x : β) ^ 2 β€ f y β§ f y β€ (f x : β) ^ 3)) :=
sorry
|
putnam_1964_b5
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals sequences normedtype topology.
From mathcomp Require Import classical_sets.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
Variable R : realType.
Theorem putnam_1964_b5
(a b : nat -> nat)
(ha : increasing_seq a /\ (forall n : nat, lt 0 (a n)))
(hb : b 0%nat = a 0%nat /\ forall n : nat, b n.+1 = lcmn (b n) (a (n.+1)))
(f : nat -> R := fun N => \sum_(0 <= n < N) 1/((b n)%:R))
: exists L : R, f @ \oo --> L.
Proof. Admitted.
|
import Mathlib
open Set Function Filter Topology
/--
Let $a_n$ be a strictly monotonic increasing sequence of positive integers. Let $b_n$ be the least common multiple of $a_1, a_2, \dots , a_n$. Prove that $\sum_{n=1}^{\infty} 1/b_n$ converges.
-/
theorem putnam_1964_b5
(a b : β β β)
(ha : StrictMono a β§ β n : β, a n > 0)
(hb : b 0 = a 0 β§ β n : β, b (n + 1) = lcm (b n) (a (n + 1)))
: (β L : β, Tendsto (fun N β¦ β n β Finset.range N, (1 : β) / b n) atTop (π L)) :=
sorry
|
putnam_2010_a6
|
Require Import Reals Coquelicot.Coquelicot.
Theorem putnam_2010_a6
(f: R -> R)
(hf : (forall (x: R), x >= 0 -> (forall (y: R), x < y -> f x > f y) /\ continuity_pt f x) /\ is_lim f p_infty 0)
: ~ ex_finite_lim (fun nInc => RInt (fun x => (f x - f (x + 1)) / f x) 0 nInc) p_infty.
Proof. Admitted.
|
import Mathlib
open Filter Topology Set
-- Note: uses (β β β) instead of (Ici 0 β β)
/--
Let $f:[0,\infty)\to \mathbb{R}$ be a strictly decreasing continuous function
such that $\lim_{x\to\infty} f(x) = 0$. Prove that
$\int_0^\infty \frac{f(x)-f(x+1)}{f(x)}\,dx$ diverges.
-/
theorem putnam_2010_a6
(f : β β β)
(hf : (StrictAntiOn f (Ici 0) β§ ContinuousOn f (Ici 0) β§ Tendsto f atTop (π 0)))
: Β¬β y : β, Tendsto (fun z : β => β« x in Ico 0 z, ((f x) - f (x + 1))/(f x)) atTop (π y) :=
sorry
|
putnam_1989_a2
|
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1989_a2_solution (a b: R) := (exp (pow (a*b) 2) - 1)/(a * b).
Theorem putnam_1989_a2
(a b: R)
(abpos : a > 0 /\ b > 0)
(f : R -> R -> R := fun x y => Rmax (pow (b*x) 2) (pow (a*y) 2))
: RInt (fun x => (RInt (fun y => exp (f x y)) 0 b)) 0 a = putnam_1989_a2_solution a b.
Proof. Admitted.
|
import Mathlib
noncomputable abbrev putnam_1989_a2_solution : β β β β β := sorry
-- (fun a b : β => (Real.exp (a ^ 2 * b ^ 2) - 1) / (a * b))
/--
Evaluate $\int_0^a \int_0^b e^{\max\{b^2x^2,a^2y^2\}}\,dy\,dx$ where $a$ and $b$ are positive.
-/
theorem putnam_1989_a2
(a b : β)
(abpos : a > 0 β§ b > 0)
: β« x in Set.Ioo 0 a, β« y in Set.Ioo 0 b, Real.exp (max (b ^ 2 * x ^ 2) (a ^ 2 * y ^ 2)) = putnam_1989_a2_solution a b :=
sorry
|
putnam_2015_a5
|
Require Import Nat Reals Arith Znumtheory Ensembles Finite_sets.
Open Scope nat_scope.
Theorem putnam_2015_a5
(q : nat)
(Nq : nat)
(qodd : odd q = true )
(qpos : q > 0)
(hNq : cardinal nat (fun a : nat => and (Rle R0 (INR a)) (Rle (INR a) (Rdiv (INR q) (INR 4))) /\ gcd a q = 1) Nq)
: odd Nq = true <-> (exists (p k : nat), q = p^k /\ k > 0 /\ prime (Z.of_nat p) /\ (p mod 8 = 5 \/ p mod 8 = 7)).
Proof. Admitted.
|
import Mathlib
/--
Let $q$ be an odd positive integer, and let $N_q$ denote the number of integers $a$ such that $0<a<q/4$ and $\gcd(a,q)=1$. Show that $N_q$ is odd if and only if $q$ is of the form $p^k$ with $k$ a positive integer and $p$ a prime congruent to $5$ or $7$ modulo $8$.
-/
theorem putnam_2015_a5
(q : β)
(Nq : β)
(qodd : Odd q)
(qpos : q > 0)
(hNq : Nq = {a : β | 0 < a β§ a < (q : β) / 4 β§ Nat.gcd a q = 1}.encard)
: Odd Nq β β p k : β, q = p ^ k β§ k > 0 β§ p.Prime β§ (p % 8 = 5 β¨ p % 8 = 7) :=
sorry
|
putnam_2014_a4
|
From mathcomp.analysis Require Import probability.
From mathcomp Require Import all_ssreflect.
From mathcomp Require Import ssralg poly ssrnum ssrint interval finmap.
From mathcomp Require Import mathcomp_extra boolp classical_sets functions.
From mathcomp Require Import cardinality fsbigop.
From HB Require Import structures.
From mathcomp.analysis Require Import exp numfun lebesgue_measure lebesgue_integral.
From mathcomp.analysis Require Import reals ereal signed topology normedtype sequences esum measure.
From mathcomp.analysis Require Import exp numfun lebesgue_measure lebesgue_integral kernel.
Context d (T : measurableType d) (R : realType) (P : probability T R).
Definition putnam_2014_a4_solution : R := 1 / 3.
Open Scope ring_scope.
Theorem putnam_2014_a4
(X : {RV P >-> R})
(ed := @expectation _ _ _ P X = 1%:E)
(ed2 := @expectation _ _ _ P (X * X) = 2%:E)
(ed3 := @expectation _ _ _ P (X * X * X) = 5%:E)
: (forall (P : probability T R), putnam_2014_a4_solution <= (pmf X 0)) /\
(exists (P : probability T R), putnam_2014_a4_solution = (pmf X 0)).
Proof. Admitted.
|
import Mathlib
open Topology Filter Nat
noncomputable abbrev putnam_2014_a4_solution : β := sorry
-- 1 / 3
/--
Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E\left[X\right]=1$, $E\left[X^2\right]=2$, and $E\left[X^3\right]=5$. (Here $E\left[Y\right]$ denotes the expectation of the random variable $Y$.) Determine the smallest possible value of the probability of the event $X=0$.
-/
theorem putnam_2014_a4
(Xrandvar : (β β β) β Prop)
(hXrandvar : Xrandvar = (fun X : β β β => (β n : β, X n β Set.Icc 0 1) β§ β' n : β, X n = 1))
(E : (β β β) β (β β β) β β)
(hE : E = (fun (X : β β β) (f : β β β) => β' n : β, f n * X n))
: sInf {X0 : β | β X : β β β, Xrandvar X β§ E X (fun x : β => x) = 1 β§ E X (fun x : β => x ^ 2) = 2 β§ E X (fun x : β => x ^ 3) = 5 β§ X0 = X 0} = putnam_2014_a4_solution :=
sorry
|
putnam_1971_b1
|
Require Import Ensembles RelationClasses.
Theorem putnam_1971_b1
(S : Type)
(op : S -> S -> S)
(hself : forall x : S, op x x = x)
(h2 : forall x y z : S, op (op x y) z = op (op y z) x)
: (forall x y z : S, op (op x y) z = op x (op y z)) /\ (forall x y : S, op x y = op y x).
Proof. Admitted.
|
import Mathlib
open Set MvPolynomial
/--
Let $S$ be a set and let $\cdot$ be a binary operation on $S$ satisfying the two following laws: (1) for all $x$ in $S$, $x = x \cdot x$, (2) for all $x,y,z$ in $S$, $(x \cdot y) \cdot z) = (y \cdot z) \cdot x$. Show that $\cdot$ is associative and commutative.
-/
theorem putnam_1971_b1
(S : Type*) [Mul S]
(hself : β x : S, x * x = x)
(h2 : β x y z : S, (x * y) * z = (y * z) * x)
: (β x y z : S, (x * y) * z = x * (y * z)) β§ (β x y : S, x * y = y * x) :=
sorry
|
putnam_1982_a5
|
Require Import Reals.
Open Scope R.
Theorem putnam_1982_a5
(a b c d: nat)
(hpos : Nat.lt 0 a /\ Nat.lt 0 b /\ Nat.lt 0 c /\ Nat.lt 0 d)
(habcd : Nat.le (Nat.add a c) 1982 /\ INR a / INR b + INR c / INR d < 1)
: 1 - INR a / INR b - INR c / INR d > 1/pow 1983 3.
Proof. Admitted.
|
import Mathlib
/--
Let $a, b, c, d$ be positive integers satisfying $a + c \leq 1982$ and $\frac{a}{b} + \frac{c}{d} < 1$. Prove that $1 - \frac{a}{b} - \frac{c}{d} > \frac{1}{1983^3}$.
-/
theorem putnam_1982_a5
(a b c d : β€)
(hpos : a > 0 β§ b > 0 β§ c > 0 β§ d > 0)
(hac : a + c β€ 1982)
(hfrac : (a : β) / b + (c : β) / d < 1)
: (1 - (a : β) / b - (c : β) / d > 1 / 1983 ^ 3) :=
sorry
|
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