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[
{
"content": "## Task B-1.3.\n\nA ship traveling along a river has covered $24 \\mathrm{~km}$ upstream and $28 \\mathrm{~km}$ downstream. For this journey, it took half an hour less than for traveling $30 \\mathrm{~km}$ upstream and $21 \\mathrm{~km}$ downstream, or half an hour more than for traveling $15 \\ma... | 3,117 | open-r1/OpenR1-Math-220k |
[
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"content": "3. (6 points) A construction company was building a tunnel. When $\\frac{1}{3}$ of the tunnel was completed at the original speed, they started using new equipment, which increased the construction speed by $20 \\%$ and reduced the working hours to $80 \\%$ of the original. As a result, it took a ... | 2,708 | open-r1/OpenR1-Math-220k |
[
{
"content": "4. Given the three sides of an obtuse triangle are 3, $4, x$, then the range of values for $x$ is ( ).\n(A) $1<x<7$.\n(B) $5 \\ll x<7$.\n(C) $1<x<\\sqrt{7}$.\n(D) $5<x<7$ or $1<x<\\sqrt{7}$.",
"role": "user"
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"content": "<think>\nOkay, so I have this problem here where I need to fi... | 2,101 | open-r1/OpenR1-Math-220k |
[
{
"content": "1. Solve the equation: $\\frac{8 x+13}{3}=\\frac{19-12 x}{2}-\\left(16 x-\\frac{7-4 x}{6}\\right)$.",
"role": "user"
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"content": "<think>\nOkay, so I need to solve this equation: (8x + 13)/3 = (19 - 12x)/2 - (16x - (7 - 4x)/6). Hmm, that looks a bit complicated with all the fractio... | 1,749 | open-r1/OpenR1-Math-220k |
[
{
"content": "Task B-4.2. Let $n$ be the number obtained by writing 2013 zeros between every two digits of the number 14641. Determine all solutions of the equation $x^{4}=n$ in the set $\\mathbb{C}$.",
"role": "user"
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"content": "<think>\nOkay, let's see. I need to find all complex solutions to... | 3,739 | open-r1/OpenR1-Math-220k |
[
{
"content": "## Problem Statement\n\nCalculate the definite integral:\n\n$$\n\\int_{0}^{3 / 2} \\frac{x^{2} \\cdot d x}{\\sqrt{9-x^{2}}}\n$$",
"role": "user"
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"content": "<think>\nOkay, so I need to calculate the definite integral of x squared over the square root of (9 minus x squared) from 0 ... | 3,425 | open-r1/OpenR1-Math-220k |
[
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"content": "Example 1 (Question from the 13th \"Hope Cup\" Invitational Competition) The real roots of the equations $x^{5}+x+1=0$ and $x+\\sqrt[5]{x}+1=0$ are $\\alpha, \\beta$ respectively, then $\\alpha+\\beta$ equals ( ).\nA. -1\nB. $-\\frac{1}{2}$\nC. $\\frac{1}{2}$\nD. 1",
"role": "user"
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{
... | 3,460 | open-r1/OpenR1-Math-220k |
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"content": "Example 5 Given that $x_{1}, x_{2}, \\cdots, x_{10}$ are all positive integers, and $x_{1}+x_{2}+\\cdots+x_{10}=2005$, find the maximum and minimum values of $x_{1}^{2}+x_{2}^{2}+\\cdots+x_{10}^{2}$.",
"role": "user"
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"content": "<think>\nOkay, let's see. I need to find the maximum... | 2,700 | open-r1/OpenR1-Math-220k |
[
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"content": "1. If the set\n$$\nA=\\{1,2, \\cdots, 10\\}, B=\\{1,2,3,4\\},\n$$\n$C$ is a subset of $A$, and $C \\cap B \\neq \\varnothing$, then the number of such subsets $C$ is $(\\quad)$.\n(A) 256\n(B) 959\n(C) 960\n(D) 961",
"role": "user"
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"content": "<think>\nOkay, so I need to figure out... | 2,364 | open-r1/OpenR1-Math-220k |
[
{
"content": "Augusto has a wire that is $10 \\mathrm{~m}$ long. He makes a cut at a point on the wire, obtaining two pieces. One piece has a length of $x$ and the other has a length of $10-x$ as shown in the figure below:\n\n![](https://cdn.mathpix.com/cropped/2024_05_01_d02c2755ad3373bde08ag-05.jpg?height=645... | 3,165 | open-r1/OpenR1-Math-220k |
[
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"content": "29. Choose any three numbers from $a, b, c, d, e$ and find their sum, exactly obtaining the ten different numbers $7,11,13,14,19,21,22,25,26,28$. Then $a+b+c+d+e=$ ( i). MATHEMATICE YoUTH CLUE A. 25\nB. 31\nC. 37\nD. 43",
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"content": "<think>\nOkay, so I need to figu... | 3,220 | open-r1/OpenR1-Math-220k |
[
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"content": "Problem 3. In the school, there are 50 teachers, of whom 29 drink coffee, 28 drink tea, and 16 do not drink either coffee or tea. How many teachers drink only coffee, and how many drink only tea?",
"role": "user"
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"content": "<think>\nOkay, let's see. There's a problem about teache... | 1,327 | open-r1/OpenR1-Math-220k |
[
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"content": "12.180. A side of the triangle is equal to 15, the sum of the other two sides is 27. Find the cosine of the angle opposite the given side, if the radius of the inscribed circle in the triangle is 4.",
"role": "user"
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"content": "<think>\nOkay, let's see. The problem says that one s... | 2,612 | open-r1/OpenR1-Math-220k |
[
{
"content": "## Problem Statement\n\nCalculate the limit of the numerical sequence:\n\n$\\lim _{n \\rightarrow \\infty} \\frac{(n+1)^{4}-(n-1)^{4}}{(n+1)^{3}+(n-1)^{3}}$",
"role": "user"
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"content": "<think>\nOkay, so I need to find the limit as n approaches infinity of the expression [(n+1)^4 ... | 3,640 | open-r1/OpenR1-Math-220k |
[
{
"content": "\\section*{Problem 5 - 071225}\n\nAll ordered pairs of real numbers \\((x, y)\\) are to be determined for which the system of equations\n\n\\[\n\\begin{aligned}\nx \\cdot\\left(a x^{2}+b y^{2}-a\\right) & =0 \\\\\ny \\cdot\\left(a x^{2}+b y^{2}-b\\right) & =0\n\\end{aligned}\n\\]\n\nis satisfied. ... | 3,647 | open-r1/OpenR1-Math-220k |
[
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"content": "40. The sum of three consecutive odd numbers is equal to the fourth power of a single-digit number. Find all such triples of numbers.",
"role": "user"
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"content": "<think>\nOkay, let's see. The problem says that the sum of three consecutive odd numbers is equal to the fourth power ... | 2,732 | open-r1/OpenR1-Math-220k |
[
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"content": "Two is $10 \\%$ of $x$ and $20 \\%$ of $y$. What is $x - y$?\n$(\\mathrm {A}) \\ 1 \\qquad (\\mathrm {B}) \\ 2 \\qquad (\\mathrm {C})\\ 5 \\qquad (\\mathrm {D}) \\ 10 \\qquad (\\mathrm {E})\\ 20$",
"role": "user"
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"content": "<think>\nOkay, let's see here. The problem says that two... | 1,217 | open-r1/OpenR1-Math-220k |
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"content": "1. (6 points) Today is January 31, 2015, welcome to the 2015 \"Spring Cup\" Final. The calculation result of the expression $\\frac{\\frac{2015}{1}+\\frac{2015}{0.31}}{1+0.31}$ is",
"role": "user"
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"content": "<think>\nOkay, so I need to calculate this expression: (2015/1 + 2015/0.... | 2,430 | open-r1/OpenR1-Math-220k |
[
{
"content": "Consider a regular hexagon with an incircle. What is the ratio of the area inside the incircle to the area of the hexagon?\n",
"role": "user"
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"content": "<think>\nOkay, so I need to find the ratio of the area inside the incircle of a regular hexagon to the area of the hexagon itse... | 2,554 | open-r1/OpenR1-Math-220k |
[
{
"content": "30. Find the remainder when the 2018-digit number $\\underbrace{\\overline{55 \\cdots}}_{2018 \\text { 555 }}$ is divided by 13.",
"role": "user"
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"content": "<think>\nOkay, so I need to find the remainder when this huge 2018-digit number made up entirely of 5s is divided by 13. Hm... | 2,879 | open-r1/OpenR1-Math-220k |
[
{
"content": "1. A line $l$ intersects a hyperbola $c$, then the maximum number of intersection points is ( ).\nA. 1\nB. 2\nC. 3\nD. 4",
"role": "user"
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"content": "<think>\nOkay, so I need to figure out the maximum number of points where a line can intersect a hyperbola. The options are 1, 2, 3... | 2,663 | open-r1/OpenR1-Math-220k |
[
{
"content": "1. Let the universal set be the set of real numbers. If $A=\\{x \\mid \\sqrt{x-2} \\leqslant 0\\}, B=\\left\\{x \\mid 10^{x^{2}-2}=10^{x}\\right\\}$, then $A \\cap \\bar{B}$ is\nA. $\\{2\\}$\nB. $\\{-1\\}$\nC. $\\{x \\mid x \\leqslant 2\\}$\nD. $\\varnothing$",
"role": "user"
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{
"con... | 1,805 | open-r1/OpenR1-Math-220k |
[
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"content": "7.242. $\\left(16 \\cdot 5^{2 x-1}-2 \\cdot 5^{x-1}-0.048\\right) \\lg \\left(x^{3}+2 x+1\\right)=0$.",
"role": "user"
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"content": "<think>\nOkay, let's see. I need to solve the equation: \n\n\\[16 \\cdot 5^{2x-1} - 2 \\cdot 5^{x-1} - 0.048 = 0\\]\nmultiplied by \n\\[\\lg(x^3 + 2x ... | 3,499 | open-r1/OpenR1-Math-220k |
[
{
"content": "In order for Mateen to walk a kilometer (1000m) in his rectangular backyard, he must walk the length 25 times or walk its perimeter 10 times. What is the area of Mateen's backyard in square meters?\n$\\text{(A)}\\ 40 \\qquad \\text{(B)}\\ 200 \\qquad \\text{(C)}\\ 400 \\qquad \\text{(D)}\\ 500 \\q... | 1,849 | open-r1/OpenR1-Math-220k |
[
{
"content": "The knights in a certain kingdom come in two colors. $\\frac{2}{7}$ of them are red, and the rest are blue. Furthermore, $\\frac{1}{6}$ of the knights are magical, and the fraction of red knights who are magical is $2$ times the fraction of blue knights who are magical. What fraction of red knight... | 1,711 | open-r1/OpenR1-Math-220k |
[
{
"content": "A father wants to divide his property among his children: first, he gives 1000 yuan and one-tenth of the remaining property to the eldest child, then 2000 yuan and one-tenth of the remaining property to the second child, then 3000 yuan and one-tenth of the remaining property to the third child, an... | 3,805 | open-r1/OpenR1-Math-220k |
[
{
"content": "Example 11 Let $x>0, y>0, \\sqrt{x}(\\sqrt{x}+2 \\sqrt{y})$ $=\\sqrt{y}(6 \\sqrt{x}+5 \\sqrt{y})$. Find the value of $\\frac{x+\\sqrt{x y}-y}{2 x+\\sqrt{x y}+3 y}$.",
"role": "user"
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"content": "<think>\nOkay, so I need to solve this problem where x and y are both positive numbers.... | 2,116 | open-r1/OpenR1-Math-220k |
[
{
"content": "Let's determine all the triples of numbers $(x, y, m)$ for which\n\n$$\n-2 x + 3 y = 2 m, \\quad x - 5 y = -11\n$$\n\nand $x$ is a negative integer, $y$ is a positive integer, and $m$ is a real number.",
"role": "user"
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{
"content": "<think>\nOkay, so I need to find all triples (x, y... | 2,008 | open-r1/OpenR1-Math-220k |
[
{
"content": "Let's determine $m$ such that the expression\n\n$$\nm x^{2}+(m-1) x+m-1\n$$\n\nis negative for all values of $x$.\n\n---\n\nDetermine $m$ so that the expression\n\n$$\nm x^{2}+(m-1) x+m-1\n$$\n\nis negative for all values of $x$.",
"role": "user"
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{
"content": "<think>\nOkay, so I ne... | 3,720 | open-r1/OpenR1-Math-220k |
[
{
"content": "Example 1 The range of the function $y=-x^{2}-2 x+3(-5 \\leqslant x \\leqslant 0)$ is $(\\quad)$.\n(A) $(-\\infty, 4]$\n(B) $[3,12]$\n(C) $[-12,4]$\n(D) $[4,12]$",
"role": "user"
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{
"content": "<think>\nOkay, let's see. I need to find the range of the function y = -x² - 2x + 3 when x... | 1,407 | open-r1/OpenR1-Math-220k |
[
{
"content": "All three vertices of $\\bigtriangleup ABC$ lie on the parabola defined by $y=x^2$, with $A$ at the origin and $\\overline{BC}$ parallel to the $x$-axis. The area of the triangle is $64$. What is the length of $BC$? \n$\\textbf{(A)}\\ 4\\qquad\\textbf{(B)}\\ 6\\qquad\\textbf{(C)}\\ 8\\qquad\\text... | 2,214 | open-r1/OpenR1-Math-220k |
[
{
"content": "2. As shown in Figure 1, the side length of rhombus $A B C D$ is $a$, and $O$ is a point on the diagonal $A C$, with $O A=a, O B=$ $O C=O D=1$. Then $a$ equals ( ).\n(A) $\\frac{\\sqrt{5}+1}{2}$\n(B) $\\frac{\\sqrt{5}-1}{2}$\n(C) 1\n(D) 2",
"role": "user"
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"content": "<think>\nOkay... | 3,799 | open-r1/OpenR1-Math-220k |
[
{
"content": "V-2 If one side of the rectangle is reduced by $3 \\mathrm{~cm}$, and the other side is reduced by $2 \\mathrm{~cm}$, we get a square whose area is $21 \\mathrm{~cm}^{2}$ less than the area of the rectangle. Calculate the dimensions of the rectangle.\n\n![](https://cdn.mathpix.com/cropped/2024_06_... | 2,845 | open-r1/OpenR1-Math-220k |
[
{
"content": "18. (3 points) Li Shuang rides a bike at a speed of 320 meters per minute from location $A$ to location $B$. On the way, due to a bicycle malfunction, he pushes the bike and walks for 5 minutes to a place 1800 meters from $B$ to repair the bike. After 15 minutes, he continues towards $B$ at 1.5 ti... | 3,551 | open-r1/OpenR1-Math-220k |
[
{
"content": "53. How many four-digit numbers contain at least one even digit?",
"role": "user"
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"content": "<think>\nOkay, so I need to figure out how many four-digit numbers contain at least one even digit. Hmm, let's see. First, let me make sure I understand the problem correctly. A four-digi... | 3,395 | open-r1/OpenR1-Math-220k |
[
{
"content": "## Problem 1\n\nPerform the calculations:\n\na) $7 \\cdot 147 - 7 \\cdot 47$ (1p)\n\nb) $(2+4+6+8+\\cdots+50)-(1+3+5+7+\\cdots+49)$ (2p)\n\nc) $10 \\cdot 9^{2} : 3^{2} - 3^{4} \\quad(2 \\text{p})$\n\nd) $(\\overline{a b} + \\overline{b c} + \\overline{c a}) : (a + b + c) \\quad$ (2p)",
"role":... | 2,542 | open-r1/OpenR1-Math-220k |
[
{
"content": "1. The curve $(x+2 y+a)\\left(x^{2}-y^{2}\\right)=0$ represents three straight lines intersecting at one point on the plane if and only if\nA. $a=0$\nB. $a=1$\nC. $a=-1$\nD. $a \\in \\mathbf{R}$",
"role": "user"
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"content": "<think>\nAlright, so I have this problem here: The curve ... | 1,748 | open-r1/OpenR1-Math-220k |
[
{
"content": "9.27 In the metro train at the initial stop, 100 passengers entered. How many ways are there to distribute the exit of all these passengers at the next 16 stops of the train?",
"role": "user"
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"content": "<think>\nOkay, so the problem is: At the initial stop, 100 passengers board a... | 2,828 | open-r1/OpenR1-Math-220k |
[
{
"content": "2. In the complex plane, there are 7 points corresponding to the 7 roots of the equation $x^{7}=$ $-1+\\sqrt{3} i$. Among the four quadrants where these 7 points are located, only 1 point is in ( ).\n(A) the I quadrant\n(B) the II quadrant\n(C) the III quadrant\n(D) the IV quadrant",
"role": "... | 3,653 | open-r1/OpenR1-Math-220k |
[
{
"content": "Of the 36 students in Richelle's class, 12 prefer chocolate pie, 8 prefer apple, and 6 prefer blueberry. Half of the remaining students prefer cherry pie and half prefer lemon. For Richelle's pie graph showing this data, how many degrees should she use for cherry pie? \n\n$ \\text{(A)}\\ 10\\qquad... | 1,286 | open-r1/OpenR1-Math-220k |
[
{
"content": "Father played chess with uncle. For a won game, the winner received 8 crowns from the opponent, and for a draw, nobody got anything. Uncle won four times, there were five draws, and in the end, father earned 24 crowns.\n\nHow many games did father play with uncle?\n\n(M. Volfová)",
"role": "us... | 1,264 | open-r1/OpenR1-Math-220k |
[
{
"content": "4. Let $A$ and $B$ be $n$-digit numbers, where $n$ is odd, which give the same remainder $r \\neq 0$ when divided by $k$. Find at least one number $k$, which does not depend on $n$, such that the number $C$, obtained by appending the digits of $A$ and $B$, is divisible by $k$.",
"role": "user"... | 2,686 | open-r1/OpenR1-Math-220k |
[
{
"content": "5. Given positive real numbers $a$ and $b$ satisfy $a+b=1$, then $M=$ $\\sqrt{1+a^{2}}+\\sqrt{1+2 b}$ the integer part is",
"role": "user"
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{
"content": "<think>\nOkay, so I need to find the integer part of M, which is defined as the sum of two square roots: √(1 + a²) and √(1 + 2b. T... | 3,532 | open-r1/OpenR1-Math-220k |
[
{
"content": "8. If $n$ is a natural number less than 50, find all values of $n$ such that the values of the algebraic expressions $4n+5$ and $7n+6$ have a common divisor greater than 1.",
"role": "user"
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{
"content": "<think>\nOkay, so I need to find all natural numbers n less than 50 where the e... | 2,600 | open-r1/OpenR1-Math-220k |
[
{
"content": "9. (16 points) Let the real number $t \\in [0, \\pi]$. If the equation $\\cos (x+t)=1-\\cos x$ has a solution for $x$, find the range of values for $t$.\n\n",
"role": "user"
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"content": "<think>\nOkay, so I need to find the range of values for \\( t \\) in the interval \\([0, \\pi]... | 3,574 | open-r1/OpenR1-Math-220k |
[
{
"content": "## Task 2 - 080522\n\nIn a warehouse, there are three times as many kilograms of wheat as in a second one. After removing $85000 \\mathrm{~kg}$ from the first and $5000 \\mathrm{~kg}$ from the second, the stocks were equal.\n\nHow many tons of wheat were in the first and how many in the second war... | 1,435 | open-r1/OpenR1-Math-220k |
[
{
"content": "$17 \\cdot 27$ As shown in the figure, in $\\triangle A B C$, $\\angle A=42^{\\circ}$. If the trisectors of $\\angle B$ and $\\angle C$ intersect at $D, E$. Then the degree measure of $\\angle B D C$ is\n(A) $67^{\\circ}$.\n(B) $84^{\\circ}$.\n(C) $88^{\\circ}$.\n(D) $110^{\\circ}$.\n(China Beijin... | 3,599 | open-r1/OpenR1-Math-220k |
[
{
"content": "2. Given $0<x<1$. Simplify\n$$\n=\\quad \\sqrt{\\left(x-\\frac{1}{x}\\right)^{2}+4}-\\sqrt{\\left(x+\\frac{1}{x}\\right)^{2}-4}\n$$",
"role": "user"
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"content": "<think>\nOkay, let's see. I need to simplify the expression √[(x - 1/x)² + 4] - √[(x + 1/x)² - 4] where x is between 0 a... | 2,523 | open-r1/OpenR1-Math-220k |
[
{
"content": "3. There are ten small balls of the same size, five of which are red and five are white. Now, these ten balls are arranged in a row arbitrarily, and numbered from left to right as $1,2, \\cdots, 10$. Then the number of arrangements where the sum of the numbers of the red balls is greater than the ... | 3,515 | open-r1/OpenR1-Math-220k |
[
{
"content": "149. Two equally matched opponents are playing chess. Find the most probable number of wins for any chess player if $2 N$ decisive (without draws) games will be played.",
"role": "user"
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"content": "<think>\nOkay, so we have two equally matched opponents playing chess, and they're ... | 1,739 | open-r1/OpenR1-Math-220k |
[
{
"content": "|\n\nThere are 85 balloons in the room - red and blue. It is known that: 1) at least one of the balloons is red; 2) in every arbitrarily chosen pair of balloons, at least one is blue. How many red balloons are in the room?\n\n#",
"role": "user"
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"content": "<think>\nOkay, let's try... | 1,300 | open-r1/OpenR1-Math-220k |
[
{
"content": "$7 \\cdot 74$ There are several locks, and now six people each hold a part of the keys. It is known that any two people trying to open the locks together will have exactly one lock that they cannot open, while any three people can open all the locks. How many locks are there at least?\n\n---\n\nNo... | 2,303 | open-r1/OpenR1-Math-220k |
[
{
"content": "22. A year is called a leap year if it is either divisible by 4 but not divisible by 100 , or divisible by 400 . Hence, the years 2000, 2004 and 2400 are leap years while the years 2006, 2100 and 2200 are not. Find the number of leap years between 2000 and 4000 inclusive.",
"role": "user"
},... | 2,554 | open-r1/OpenR1-Math-220k |
[
{
"content": "1. a) $\\left\\{\\begin{array}{l}y=\\sqrt{x} \\\\ y=\\frac{x^{2}}{4}\\end{array}\\right.$ have the solutions $x_{1}=0, y_{1}=0$ and $x_{2}=\\sqrt[3]{16}, y_{2}=\\sqrt[3]{4}$\n\nb) $A=\\frac{1}{2}|d|$, where $\\mathrm{d}=d=\\left|\\begin{array}{lll}a & \\sqrt{a} & 1 \\\\ b & \\frac{b^{2}}{4} & 1 \\... | 2,502 | open-r1/OpenR1-Math-220k |
[
{
"content": "2. On a line, several points were marked, including points $A$ and $B$. All possible segments with endpoints at the marked points are considered. Vasya calculated that point $A$ is inside 50 of these segments, and point $B$ is inside 56 segments. How many points were marked? (The endpoints of a se... | 3,004 | open-r1/OpenR1-Math-220k |
[
{
"content": "3. (8 points) When Tongtong was calculating a division with a remainder, she mistakenly read the dividend 472 as 427, resulting in a quotient that was 5 less than the original, but the remainder was exactly the same. What is this remainder?\nA. 4\nB. 5\nC. 6\nD. 7",
"role": "user"
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{
... | 2,073 | open-r1/OpenR1-Math-220k |
[
{
"content": "3. When $\\frac{\\pi}{4} \\leqslant x \\leqslant \\frac{\\pi}{2}$, the range of $y=\\sin ^{2} x+\\sqrt{3} \\sin x \\cos x$ is $\\qquad$",
"role": "user"
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{
"content": "<think>\nOkay, so I need to find the range of the function y = sin²x + √3 sinx cosx when x is between π/4 and π/2. H... | 3,802 | open-r1/OpenR1-Math-220k |
[
{
"content": "6.7. $f(x)=4 x^{3}-\\frac{5}{x^{7}}-8 \\sqrt[6]{x^{5}}+10$.\n\n6.7. $f(x)=4 x^{3}-\\frac{5}{x^{7}}-8 \\sqrt[6]{x^{5}}+10$.",
"role": "user"
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{
"content": "<think>\nOkay, so I need to work on this function: f(x) = 4x³ - 5/x⁷ - 8 times the sixth root of x⁵ plus 10. Hmm, the problem isn... | 2,264 | open-r1/OpenR1-Math-220k |
[
{
"content": "316. $\\left\\{\\begin{array}{l}(x+y)\\left(x^{2}+y^{2}\\right)=539200 \\\\ (x-y)\\left(x^{2}-y^{2}\\right)=78400 .\\end{array}\\right.$",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to solve this system of equations:\n\n\\[\n\\begin{cases}\n(x + y)(x^2 + y^2) = 539200 \\\... | 2,731 | open-r1/OpenR1-Math-220k |
[
{
"content": "Example 1. Find the general solution of the linear differential equation\n\n$$\ny^{\\prime}+y-k x=0\n$$",
"role": "user"
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{
"content": "<think>\nOkay, so I need to find the general solution to the differential equation y' + y - kx = 0. Hmm, let's see. I remember that linear different... | 1,694 | open-r1/OpenR1-Math-220k |
[
{
"content": "What is the $21^{\\varrho}$ term of the sequence\n\n$$\n1 ; 2+3 ; 4+5+6 ; 7+8+9+10 ; 11+12+13+14+15 ; \\ldots ?\n$$",
"role": "user"
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{
"content": "<think>\nOkay, let's see. I need to find the 21st term of this sequence. The sequence is given as:\n\n1 ; 2+3 ; 4+5+6 ; 7+8+9+10 ; 11+12... | 2,467 | open-r1/OpenR1-Math-220k |
[
{
"content": "5. Solve the pair of simultaneous equations\n$(a+b)\\left(a^{2}-b^{2}\\right)=4$ and\n$$\n(a-b)\\left(a^{2}+b^{2}\\right)=\\frac{5}{2} \\text {. }\n$$",
"role": "user"
},
{
"content": "<think>\nAlright, so I need to solve these two simultaneous equations:\n\n1. \\((a + b)(a^2 - b^2) = ... | 2,687 | open-r1/OpenR1-Math-220k |
[
{
"content": "Problem 2. In two fruit stores, there were a total of $365 \\mathrm{~kg}$ of apples, which were sold at the same price. When the first store sold a certain amount of apples and received 4340 denars for it, and the second store received 8750 denars for its sold amount of apples, then in the first s... | 2,286 | open-r1/OpenR1-Math-220k |
[
{
"content": "182. Perform the operations: a) $(2+3 i)^{2} ;$ b) $(3-5 i)^{2}$; c) $(5+$ $+3 i)^{3}$",
"role": "user"
},
{
"content": "<think>\nAlright, let's tackle these complex number operations step by step. I remember that when dealing with complex numbers, the key is to treat \\(i\\) as the im... | 2,545 | open-r1/OpenR1-Math-220k |
[
{
"content": "4. The sequence $\\left\\{x_{n}\\right\\}$ is defined as follows:\n$$\nx_{1}=\\frac{2}{3}, x_{n+1}=\\frac{x_{n}}{2(2 n+1) x_{n}+1}\\left(n \\in \\mathbf{Z}_{+}\\right) \\text {. }\n$$\n\nThen $x_{1}+x_{2}+\\cdots+x_{2014}=$ $\\qquad$",
"role": "user"
},
{
"content": "<think>\nOkay, so ... | 3,473 | open-r1/OpenR1-Math-220k |
[
{
"content": "1. One car covers a distance of 120 km 18 minutes faster than the other. If the first car reduced its speed by 12 km/h, and the second car increased its speed by $10 \\%$, they would spend the same amount of time on the same distance. Find the speeds of the cars.",
"role": "user"
},
{
... | 3,578 | open-r1/OpenR1-Math-220k |
[
{
"content": "2. Given the function $y=\\frac{a-x}{x-a-1}$, the graph of its inverse function is symmetric about the point $(-1,4)$. Then the value of the real number $a$ is $\\qquad$ .",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the value of the real number 'a' given that the... | 2,970 | open-r1/OpenR1-Math-220k |
[
{
"content": "669. For which numbers $x$ does the equation $x+|x|=0$ hold?",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out for which numbers x the equation x + |x| = 0 holds. Hmm, let's start by recalling what the absolute value function does. The absolute value of x, writte... | 1,915 | open-r1/OpenR1-Math-220k |
[
{
"content": "14.18 Indicate all points $(x, y)$ in the $(x, y)$ plane that satisfy $[x]^{2}+[y]^{2}=4$.\n(7th Canadian Mathematics Competition, 1975)",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out all the points (x, y) in the plane that satisfy the equation [x]² + [y]² = 4... | 3,772 | open-r1/OpenR1-Math-220k |
[
{
"content": "Example 4 Find the integer solution of the equation $\\left[\\frac{x}{1!}\\right]+\\left[\\frac{x}{2!}\\right]+\\left[\\frac{x}{3!}\\right]+\\cdots+$ $\\left[\\frac{x}{10!}\\right]=1001$.",
"role": "user"
},
{
"content": "<think>\nAlright, so I need to find an integer solution x such t... | 3,084 | open-r1/OpenR1-Math-220k |
[
{
"content": "321. $\\sqrt{2 x+5}+\\sqrt{x-1}=8$.\n\nTranslate the text above into English, keeping the original text's line breaks and format, and output the translation result directly.\n\n321. $\\sqrt{2 x+5}+\\sqrt{x-1}=8$.",
"role": "user"
},
{
"content": "<think>\nAlright, so I have this equati... | 2,890 | open-r1/OpenR1-Math-220k |
[
{
"content": "13. (20 points) As shown in Figure 2, planes $M$ and $N$ intersect at line $l$. Points $A$ and $D$ are on line $l$, ray $DB$ is in plane $M$, and ray $DC$ is in plane $N$. Given that $\\angle BDC = \\alpha$, $\\angle BDA = \\beta$, $\\angle CDA = \\gamma$, and $\\alpha$, $\\beta$, $\\gamma$ are al... | 3,132 | open-r1/OpenR1-Math-220k |
[
{
"content": "## Problem Statement\n\nFind the coordinates of point $A$, which is equidistant from points $B$ and $C$.\n\n$A(0 ; 0 ; z)$\n\n$B(10 ; 0 ;-2)$\n\n$C(9 ;-2 ; 1)$",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the coordinates of point A, which is equidistant from point... | 2,012 | open-r1/OpenR1-Math-220k |
[
{
"content": "2、D Teacher has five vases, these five vases are arranged in a row from shortest to tallest, the height difference between adjacent vases is 2 centimeters, and the tallest vase is exactly equal to the sum of the heights of the two shortest vases, then the total height of the five vases is _ centim... | 1,570 | open-r1/OpenR1-Math-220k |
[
{
"content": "2. A helicopter and an airplane flew towards each other at the same time. By the time they met, the helicopter had flown $100 \\mathrm{~km}$ less than the airplane, and the helicopter reached the airplane's takeoff location 3 hours after the meeting. The airplane reached the helicopter's takeoff l... | 2,745 | open-r1/OpenR1-Math-220k |
[
{
"content": "Problem 10.7. The graph of the function $f(x)=\\frac{1}{12} x^{2}+a x+b$ intersects the $O x$ axis at points $A$ and $C$, and the $O y$ axis at point $B$, as shown in the figure. It turned out that for the point $T$ with coordinates $(3 ; 3)$, the condition $T A=T B=T C$ is satisfied. Find $b$.\n\... | 3,619 | open-r1/OpenR1-Math-220k |
[
{
"content": "3. The monotonic increasing interval of the function $f(x)$ $=\\frac{(x+1)^{2}}{x^{2}+1}$ is $\\qquad$",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the monotonic increasing interval of the function f(x) = (x + 1)^2 / (x² + 1). Hmm, monotonic increasing means that ... | 3,174 | open-r1/OpenR1-Math-220k |
[
{
"content": "10.024. The lengths of the parallel sides of the trapezoid are 25 and 4 cm, and the lengths of the non-parallel sides are 20 and 13 cm. Find the height of the trapezoid.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the height of a trapezoid where the parallel side... | 1,855 | open-r1/OpenR1-Math-220k |
[
{
"content": "1. Without using calculators or other computational tools, determine which of the two numbers is greater:\n\n$$\n\\frac{1+\\sqrt{5}}{2} \\text { or } \\sqrt[6]{18} ?\n$$",
"role": "user"
},
{
"content": "<think>\nAlright, so I need to figure out which is bigger: (1 + sqrt(5))/2 or the ... | 2,283 | open-r1/OpenR1-Math-220k |
[
{
"content": "6. (10 points) As shown in the figure, the area of square $A B C D$ is 840 square centimeters, $A E=E B, B F=2 F C, D F$ intersects $E C$ at $G$. Then the area of quadrilateral $A E G D$ is square centimeters.",
"role": "user"
},
{
"content": "<think>\nOkay, let's try to solve this pro... | 3,298 | open-r1/OpenR1-Math-220k |
[
{
"content": "For example, in the sequence $x_{1}, x_{2}, \\cdots, x_{n}, \\cdots$, the sum of any three consecutive terms is 20, and $x_{1}=9, x_{12}=7$. Find the value of $x_{2000}$.",
"role": "user"
},
{
"content": "<think>\nOkay, let me see. So, there's this sequence where the sum of any three c... | 3,020 | open-r1/OpenR1-Math-220k |
[
{
"content": "13.223. Point $C$ is located 12 km downstream from point $B$. A fisherman set out from point $A$, located upstream from point $B$, to point $C$. After 4 hours, he arrived at $C$, and the return trip took 6 hours. On another occasion, the fisherman used a motorboat, thereby tripling his own speed r... | 2,389 | open-r1/OpenR1-Math-220k |
[
{
"content": "Find the invertibles modulo 8, then their inverses. Do the same modulo 9.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the invertible elements modulo 8 and their inverses, and then do the same for modulo 9. Hmm, invertible elements modulo n are numbers that are re... | 2,168 | open-r1/OpenR1-Math-220k |
[
{
"content": "II. (25 points) If $2 x+y \\geqslant 1$. Try to find the minimum value of the function $u=y^{2}-2 y$ $+x^{2}+4 x$.\n\n",
"role": "user"
},
{
"content": "<think>\nOkay, so I have this problem here where I need to find the minimum value of the function u = y² - 2y + x² + 4x, given that 2... | 3,174 | open-r1/OpenR1-Math-220k |
[
{
"content": "The letter $A$ has a vertical line of symmetry and the letter $B$ does not. How many of the letters H L O R X D P E have a vertical line of symmetry?\n(A) 1\n(B) 2\n(C) 3\n(D) 4\n(E) 5",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out how many of the letters H, L... | 2,114 | open-r1/OpenR1-Math-220k |
[
{
"content": "18. Let $a_{k}$ be the coefficient of $x^{k}$ in the expansion of\n$$\n(x+1)+(x+1)^{2}+(x+1)^{3}+(x+1)^{4}+\\cdots+(x+1)^{99} \\text {. }\n$$\n\nDetermine the value of $\\left\\lfloor a_{4} / a_{3}\\right\\rfloor$.",
"role": "user"
},
{
"content": "<think>\nOkay, let's see. I need to f... | 2,503 | open-r1/OpenR1-Math-220k |
[
{
"content": "10.083. The sum of the lengths of the diagonals of a rhombus is $m$, and its area is $S$. Find the side of the rhombus.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the side of a rhombus when given the sum of the lengths of its diagonals (which is m) and its area ... | 3,246 | open-r1/OpenR1-Math-220k |
[
{
"content": "2.1 Find $f(2013)$, if for any real $x$ and $y$ the equality holds\n\n$$\nf(x-y)=f(x)+f(y)-2xy\n$$",
"role": "user"
},
{
"content": "<think>\nOkay, let's try to figure out how to find f(2013) given that for any real numbers x and y, the equation f(x - y) = f(x) + f(y) - 2xy holds. Hmm,... | 2,172 | open-r1/OpenR1-Math-220k |
[
{
"content": "Example 6.35. Find the confidence interval for estimating the mathematical expectation $a$ of a normal distribution with a reliability of 0.95, knowing the sample mean $\\bar{X}_{\\mathrm{B}}=10.43$, the sample size $n=100$, and the standard deviation $\\sigma=5$.",
"role": "user"
},
{
... | 1,550 | open-r1/OpenR1-Math-220k |
[
{
"content": "7. (10 points) Two wheels rotate, meshed with each other, around fixed axes passing through the centers of wheels $A$ and $B$. The radii of the wheels differ by a factor of three. The smaller wheel makes 30 revolutions per minute. Determine how many seconds the larger wheel spends on one revolutio... | 1,837 | open-r1/OpenR1-Math-220k |
[
{
"content": "Determine the value of $2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{2^{2}}+\\frac{1}{2^{3}}+\\frac{1}{2^{4}}\\right)$.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out the value of 2 to the power of 4 multiplied by this sum: 1 plus 1/2 plus 1 over 2 squared plus 1 over... | 1,863 | open-r1/OpenR1-Math-220k |
[
{
"content": "【Question 6】Using 2 unit squares (unit squares) can form a 2-connected square, which is commonly known as a domino. Obviously, dominoes that can coincide after translation, rotation, or symmetry transformation should be considered as the same one, so there is only one domino.\nSimilarly, the diffe... | 2,429 | open-r1/OpenR1-Math-220k |
[
{
"content": "3. It is known that for some positive coprime numbers $m$ and $n$, the numbers $m+1947 n$ and $n+1947 m$ have a common prime divisor $d>9$. Find the smallest possible value of the number $d$ under these conditions.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to solve thi... | 3,819 | open-r1/OpenR1-Math-220k |
[
{
"content": "Task B-2.1. (20 points) For complex numbers $z, w$ such that $|z|=|w|=|z-w|$, calculate $\\left(\\frac{z}{w}\\right)^{99}$.",
"role": "user"
},
{
"content": "<think>\nOkay, so I have this problem here: For complex numbers z and w where |z| = |w| = |z - w|, I need to calculate (z/w)^99.... | 2,385 | open-r1/OpenR1-Math-220k |
[
{
"content": "1. If $x-y=12$, find the value of $x^{3}-y^{3}-36 x y$.\n(1 mark)\nIf $x-y=12$, find the value of $x^{3}-y^{3}-36 x y$.\n(1 mark)",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the value of \\( x^3 - y^3 - 36xy \\) given that \\( x - y = 12 \\). Hmm, let's see. I re... | 2,170 | open-r1/OpenR1-Math-220k |
[
{
"content": "## 232. Math Puzzle $9 / 84$\n\nA motorcycle battery ( $6 \\mathrm{~V}$ ) is discharged through a lamp $6 \\mathrm{~V} / 0.5 \\mathrm{~W}$. The lamp burns for 48 hours.\n\nHow many Ah was the battery charged if we assume that the discharge current remains constant?",
"role": "user"
},
{
... | 1,234 | open-r1/OpenR1-Math-220k |
[
{
"content": "45. The sum of the interior angles of a convex $n$-sided polygon is less than $2021^{\\circ}$, then the maximum value of $n$ is",
"role": "user"
},
{
"content": "<think>\nAlright, so I need to find the maximum number of sides, n, that a convex polygon can have such that the sum of its ... | 3,495 | open-r1/OpenR1-Math-220k |
[
{
"content": "11. Given $\\frac{x y}{x+y}=2, \\frac{x z}{x+z}=3, \\frac{y z}{y+z}=4$. Find the value of $7 x+5 y-2 z$.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to solve this problem where I have three equations involving x, y, and z, and I need to find the value of 7x + 5y - 2z. Th... | 2,703 | open-r1/OpenR1-Math-220k |
[
{
"content": "8. Determine the value of the sum\n$$\n\\frac{3}{1^{2} \\cdot 2^{2}}+\\frac{5}{2^{2} \\cdot 3^{2}}+\\frac{7}{3^{2} \\cdot 4^{2}}+\\cdots+\\frac{29}{14^{2} \\cdot 15^{2}} .\n$$",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the value of this sum: 3/(1²·2²) + 5/(2²·3²... | 2,069 | open-r1/OpenR1-Math-220k |
[
{
"content": "8. Let $a>0, b>0$. Then the inequality that does not always hold is ( ).\n(A) $\\frac{2}{\\frac{1}{a}+\\frac{1}{b}} \\geqslant \\sqrt{a b}$\n(B) $\\frac{1}{a}+\\frac{1}{b} \\geqslant \\frac{4}{a+b}$\n(C) $\\sqrt{|a-b|} \\geqslant \\sqrt{a}-\\sqrt{b}$\n(D) $a^{2}+b^{2}+1>a+b$",
"role": "user"
... | 2,665 | open-r1/OpenR1-Math-220k |
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