problem_id stringlengths 17 21 | task_type stringclasses 1 value | prompt stringlengths 10 5.15k | verification_info stringlengths 20 1.24k | metadata stringclasses 1 value |
|---|---|---|---|---|
deepscaler_math_0 | verifiable_math | The operation $\otimes$ is defined for all nonzero numbers by $a \otimes b = \frac{a^{2}}{b}$. Determine $[(1 \otimes 2) \otimes 3] - [1 \otimes (2 \otimes 3)]$. | {"ground_truth": "-\\frac{2}{3}"} | {} |
deepscaler_math_1 | verifiable_math | Doug constructs a square window using $8$ equal-size panes of glass. The ratio of the height to width for each pane is $5 : 2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length of the square window? | {"ground_truth": "26"} | {} |
deepscaler_math_2 | verifiable_math | Let $P(x)$ be a polynomial of degree $3n$ such that
\begin{align*} P(0) = P(3) = \dots = P(3n) &= 2, \\ P(1) = P(4) = \dots = P(3n+1-2) &= 1, \\ P(2) = P(5) = \dots = P(3n+2-2) &= 0. \end{align*}
Also, $P(3n+1) = 730$. Determine $n$. | {"ground_truth": "1"} | {} |
deepscaler_math_3 | verifiable_math | Let $f$ be the function defined by $f(x)=ax^2-\sqrt{2}$ for some positive $a$. If $f(f(\sqrt{2}))=-\sqrt{2}$ then $a=$ | {"ground_truth": "\\frac{\\sqrt{2}}{2}"} | {} |
deepscaler_math_4 | verifiable_math | At Euclid Middle School the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are $11$ students in Mrs. Germain's class, $8$ students in Mr. Newton's class, and $9$ students in Mrs. Young's class taking the AMC $8$ this year. How many mathematics students at Euclid Middle School are taking the contest? | {"ground_truth": "28"} | {} |
deepscaler_math_5 | verifiable_math | If $991+993+995+997+999=5000-N$, then $N=$ | {"ground_truth": "25"} | {} |
deepscaler_math_6 | verifiable_math | The total in-store price for an appliance is $99.99$. A television commercial advertises the same product for three easy payments of $29.98$ and a one-time shipping and handling charge of $9.98$. How many cents are saved by buying the appliance from the television advertiser? | {"ground_truth": "7"} | {} |
deepscaler_math_7 | verifiable_math | Points $A,B,C,D,E$ and $F$ lie, in that order, on $\overline{AF}$, dividing it into five segments, each of length 1. Point $G$ is not on line $AF$. Point $H$ lies on $\overline{GD}$, and point $J$ lies on $\overline{GF}$. The line segments $\overline{HC}, \overline{JE},$ and $\overline{AG}$ are parallel. Find $HC/JE$. | {"ground_truth": "\\frac{5}{3}"} | {} |
deepscaler_math_8 | verifiable_math | During the softball season, Judy had $35$ hits. Among her hits were $1$ home run, $1$ triple and $5$ doubles. The rest of her hits were single. What percent of her hits were single? | {"ground_truth": "80\\%"} | {} |
deepscaler_math_9 | verifiable_math | The graph, $G$ of $y=\log_{10}x$ is rotated $90^{\circ}$ counter-clockwise about the origin to obtain a new graph $G'$. What is the equation for $G'$? | {"ground_truth": "10^{-x}"} | {} |
deepscaler_math_10 | verifiable_math | A set of consecutive positive integers beginning with $1$ is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is $35\frac{7}{17}$. What number was erased? | {"ground_truth": "7"} | {} |
deepscaler_math_11 | verifiable_math | A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $15$ and $25$ meters. What fraction of the yard is occupied by the flower beds?
[asy]
unitsize(2mm); defaultpen(linewidth(.8pt));
fill((0,0)--(0,5)--(5,5)--cycle,gray);
fill((25,0)--(25,5)--(20,5)--cycle,gray);
draw((0,0)--(0,5)--(25,5)--(25,0)--cycle);
draw((0,0)--(5,5));
draw((20,5)--(25,0));
[/asy] | {"ground_truth": "\\frac{1}{5}"} | {} |
deepscaler_math_12 | verifiable_math | What is the sum of all possible values of $t$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are $(\cos 40^\circ,\sin 40^\circ)$, $(\cos 60^\circ,\sin 60^\circ)$, and $(\cos t^\circ,\sin t^\circ)$ is isosceles? | {"ground_truth": "380"} | {} |
deepscaler_math_13 | verifiable_math | In the adjoining figure, points $B$ and $C$ lie on line segment $AD$, and $AB, BC$, and $CD$ are diameters of circle $O, N$, and $P$, respectively. Circles $O, N$, and $P$ all have radius $15$ and the line $AG$ is tangent to circle $P$ at $G$. If $AG$ intersects circle $N$ at points $E$ and $F$, then chord $EF$ has length | {"ground_truth": "20"} | {} |
deepscaler_math_14 | verifiable_math | The first three terms of an arithmetic progression are $x - 1, x + 1, 2x + 3$, in the order shown. The value of $x$ is: | {"ground_truth": "0"} | {} |
deepscaler_math_15 | verifiable_math | Alicia had two containers. The first was $\frac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\frac{3}{4}$ full of water. What is the ratio of the volume of the first container to the volume of the second container? | {"ground_truth": "\\frac{9}{10}"} | {} |
deepscaler_math_16 | verifiable_math | An architect is building a structure that will place vertical pillars at the vertices of regular hexagon $ABCDEF$, which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of pillars at $A$, $B$, and $C$ are $12$, $9$, and $10$ meters, respectively. What is the height, in meters, of the pillar at $E$? | {"ground_truth": "17"} | {} |
deepscaler_math_17 | verifiable_math | The points $(2,-3)$, $(4,3)$, and $(5, k/2)$ are on the same straight line. The value(s) of $k$ is (are): | {"ground_truth": "12"} | {} |
deepscaler_math_18 | verifiable_math | In a certain year the price of gasoline rose by $20\%$ during January, fell by $20\%$ during February, rose by $25\%$ during March, and fell by $x\%$ during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is $x$ | {"ground_truth": "17"} | {} |
deepscaler_math_19 | verifiable_math | Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}, AC=20,$ and $CD=30.$ Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E,$ and $AE=5.$ What is the area of quadrilateral $ABCD?$ | {"ground_truth": "360"} | {} |
deepscaler_math_20 | verifiable_math | The angle bisector of the acute angle formed at the origin by the graphs of the lines $y = x$ and $y=3x$ has equation $y=kx.$ What is $k?$ | {"ground_truth": "\\frac{1+\\sqrt{5}}{2}"} | {} |
deepscaler_math_21 | verifiable_math | A set $S$ of points in the $xy$-plane is symmetric about the origin, both coordinate axes, and the line $y=x$. If $(2,3)$ is in $S$, what is the smallest number of points in $S$? | {"ground_truth": "8"} | {} |
deepscaler_math_22 | verifiable_math | Five positive consecutive integers starting with $a$ have average $b$. What is the average of $5$ consecutive integers that start with $b$? | {"ground_truth": "$a+4$"} | {} |
deepscaler_math_23 | verifiable_math | At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. What is the product of all possible values of $N?$ | {"ground_truth": "60"} | {} |
deepscaler_math_24 | verifiable_math | Consider all 1000-element subsets of the set $\{1, 2, 3, \dots , 2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. | {"ground_truth": "2016"} | {} |
deepscaler_math_25 | verifiable_math | A man on his way to dinner shortly after $6:00$ p.m. observes that the hands of his watch form an angle of $110^{\circ}$. Returning before $7:00$ p.m. he notices that again the hands of his watch form an angle of $110^{\circ}$. The number of minutes that he has been away is: | {"ground_truth": "40"} | {} |
deepscaler_math_26 | verifiable_math | A 3x3x3 cube is made of $27$ normal dice. Each die's opposite sides sum to $7$. What is the smallest possible sum of all of the values visible on the $6$ faces of the large cube? | {"ground_truth": "90"} | {} |
deepscaler_math_27 | verifiable_math | In the multiplication problem below $A$, $B$, $C$, $D$ are different digits. What is $A+B$?
$\begin{array}{cccc} & A & B & A\\ \times & & C & D\\ \hline C & D & C & D\\ \end{array}$ | {"ground_truth": "1"} | {} |
deepscaler_math_28 | verifiable_math | Andrea and Lauren are $20$ kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lauren, and the distance between them decreasing at a rate of $1$ kilometer per minute. After $5$ minutes, Andrea stops biking because of a flat tire and waits for Lauren. After how many minutes from the time they started to bike does Lauren reach Andrea? | {"ground_truth": "65"} | {} |
deepscaler_math_29 | verifiable_math | The sum of two natural numbers is $17402$. One of the two numbers is divisible by $10$. If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers? | {"ground_truth": "14238"} | {} |
deepscaler_math_30 | verifiable_math | What is the value of $2^{0^{1^9}} + (2^0)^{1^9}$? | {"ground_truth": "2"} | {} |
deepscaler_math_31 | verifiable_math | If $a$ and $b$ are digits for which
$\begin{array}{ccc}& 2 & a\ \times & b & 3\ \hline & 6 & 9\ 9 & 2 & \ \hline 9 & 8 & 9\end{array}$
then $a+b =$ | {"ground_truth": "7"} | {} |
deepscaler_math_32 | verifiable_math | If $x, 2x+2, 3x+3, \dots$ are in geometric progression, the fourth term is: | {"ground_truth": "-13\\frac{1}{2}"} | {} |
deepscaler_math_33 | verifiable_math | At $2:15$ o'clock, the hour and minute hands of a clock form an angle of: | {"ground_truth": "22\\frac {1}{2}^{\\circ}"} | {} |
deepscaler_math_34 | verifiable_math | A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with the terms 247, 475, and 756 and end with the term 824. Let $S$ be the sum of all the terms in the sequence. What is the largest prime factor that always divides $S$? | {"ground_truth": "37"} | {} |
deepscaler_math_35 | verifiable_math | Square $EFGH$ has one vertex on each side of square $ABCD$. Point $E$ is on $AB$ with $AE=7\cdot EB$. What is the ratio of the area of $EFGH$ to the area of $ABCD$? | {"ground_truth": "\\frac{25}{32}"} | {} |
deepscaler_math_36 | verifiable_math | Chandler wants to buy a $500$ dollar mountain bike. For his birthday, his grandparents send him $50$ dollars, his aunt sends him $35$ dollars and his cousin gives him $15$ dollars. He earns $16$ dollars per week for his paper route. He will use all of his birthday money and all of the money he earns from his paper route. In how many weeks will he be able to buy the mountain bike? | {"ground_truth": "25"} | {} |
deepscaler_math_37 | verifiable_math | The harmonic mean of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4? | {"ground_truth": "\\frac{12}{7}"} | {} |
deepscaler_math_38 | verifiable_math | Julie is preparing a speech for her class. Her speech must last between one-half hour and three-quarters of an hour. The ideal rate of speech is 150 words per minute. If Julie speaks at the ideal rate, what number of words would be an appropriate length for her speech? | {"ground_truth": "5650"} | {} |
deepscaler_math_39 | verifiable_math | Let $ABC$ be an equilateral triangle. Extend side $\overline{AB}$ beyond $B$ to a point $B'$ so that $BB'=3 \cdot AB$. Similarly, extend side $\overline{BC}$ beyond $C$ to a point $C'$ so that $CC'=3 \cdot BC$, and extend side $\overline{CA}$ beyond $A$ to a point $A'$ so that $AA'=3 \cdot CA$. What is the ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$? | {"ground_truth": "16"} | {} |
deepscaler_math_40 | verifiable_math | If the following instructions are carried out by a computer, what value of \(X\) will be printed because of instruction \(5\)?
1. START \(X\) AT \(3\) AND \(S\) AT \(0\).
2. INCREASE THE VALUE OF \(X\) BY \(2\).
3. INCREASE THE VALUE OF \(S\) BY THE VALUE OF \(X\).
4. IF \(S\) IS AT LEAST \(10000\),
THEN GO TO INSTRUCTION \(5\);
OTHERWISE, GO TO INSTRUCTION \(2\).
AND PROCEED FROM THERE.
5. PRINT THE VALUE OF \(X\).
6. STOP. | {"ground_truth": "23"} | {} |
deepscaler_math_41 | verifiable_math | Letters $A, B, C,$ and $D$ represent four different digits selected from $0, 1, 2, \ldots ,9.$ If $(A+B)/(C+D)$ is an integer that is as large as possible, what is the value of $A+B$? | {"ground_truth": "17"} | {} |
deepscaler_math_42 | verifiable_math | A shopper plans to purchase an item that has a listed price greater than $\$100$ and can use any one of the three coupons. Coupon A gives $15\%$ off the listed price, Coupon B gives $\$30$ off the listed price, and Coupon C gives $25\%$ off the amount by which the listed price exceeds
$\$100$.
Let $x$ and $y$ be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or C. What is $y - x$? | {"ground_truth": "50"} | {} |
deepscaler_math_43 | verifiable_math | In $\triangle ABC$, $\angle ABC=45^\circ$. Point $D$ is on $\overline{BC}$ so that $2 \cdot BD=CD$ and $\angle DAB=15^\circ$. Find $\angle ACB.$ | {"ground_truth": "75^\\circ"} | {} |
deepscaler_math_44 | verifiable_math | Mary's top book shelf holds five books with the following widths, in centimeters: $6$, $\dfrac{1}{2}$, $1$, $2.5$, and $10$. What is the average book width, in centimeters? | {"ground_truth": "4"} | {} |
deepscaler_math_45 | verifiable_math | The sum of the greatest integer less than or equal to $x$ and the least integer greater than or equal to $x$ is $5$. The solution set for $x$ is | {"ground_truth": "\\{x \\mid 2 < x < 3\\}"} | {} |
deepscaler_math_46 | verifiable_math | A powderman set a fuse for a blast to take place in $30$ seconds. He ran away at a rate of $8$ yards per second. Sound travels at the rate of $1080$ feet per second. When the powderman heard the blast, he had run approximately: | {"ground_truth": "245 yd."} | {} |
deepscaler_math_47 | verifiable_math | A student council must select a two-person welcoming committee and a three-person planning committee from among its members. There are exactly $10$ ways to select a two-person team for the welcoming committee. It is possible for students to serve on both committees. In how many different ways can a three-person planning committee be selected? | {"ground_truth": "10"} | {} |
deepscaler_math_48 | verifiable_math | Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be $2$ : $1$ ? | {"ground_truth": "4"} | {} |
deepscaler_math_49 | verifiable_math | A square piece of paper, 4 inches on a side, is folded in half vertically. Both layers are then cut in half parallel to the fold. Three new rectangles are formed, a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle? | {"ground_truth": "\\frac{5}{6}"} | {} |
deepscaler_math_50 | verifiable_math | Positive integers $a$ and $b$ are such that the graphs of $y=ax+5$ and $y=3x+b$ intersect the $x$-axis at the same point. What is the sum of all possible $x$-coordinates of these points of intersection? | {"ground_truth": "-8"} | {} |
deepscaler_math_51 | verifiable_math | Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ = 48$. Let $R$ be the midpoint of $\overline{PQ}$. What is $AR+BR+CR+DR$? | {"ground_truth": "192"} | {} |
deepscaler_math_52 | verifiable_math | Medians $AD$ and $CE$ of $\triangle ABC$ intersect in $M$. The midpoint of $AE$ is $N$.
Let the area of $\triangle MNE$ be $k$ times the area of $\triangle ABC$. Then $k$ equals: | {"ground_truth": "\\frac{1}{6}"} | {} |
deepscaler_math_53 | verifiable_math | Find the minimum value of $\sqrt{x^2+y^2}$ if $5x+12y=60$. | {"ground_truth": "\\frac{60}{13}"} | {} |
deepscaler_math_54 | verifiable_math | On average, for every 4 sports cars sold at the local dealership, 7 sedans are sold. The dealership predicts that it will sell 28 sports cars next month. How many sedans does it expect to sell? | {"ground_truth": "49"} | {} |
deepscaler_math_55 | verifiable_math | Two fair dice, each with at least $6$ faces are rolled. On each face of each die is printed a distinct integer from $1$ to the number of faces on that die, inclusive. The probability of rolling a sum of $7$ is $\frac34$ of the probability of rolling a sum of $10,$ and the probability of rolling a sum of $12$ is $\frac{1}{12}$. What is the least possible number of faces on the two dice combined? | {"ground_truth": "17"} | {} |
deepscaler_math_56 | verifiable_math | Two long cylindrical tubes of the same length but different diameters lie parallel to each other on a flat surface. The larger tube has radius $72$ and rolls along the surface toward the smaller tube, which has radius $24$. It rolls over the smaller tube and continues rolling along the flat surface until it comes to rest on the same point of its circumference as it started, having made one complete revolution. If the smaller tube never moves, and the rolling occurs with no slipping, the larger tube ends up a distance $x$ from where it starts. The distance $x$ can be expressed in the form $a\pi+b\sqrt{c},$ where $a,$ $b,$ and $c$ are integers and $c$ is not divisible by the square of any prime. Find $a+b+c.$ | {"ground_truth": "312"} | {} |
deepscaler_math_57 | verifiable_math | The number $2.5252525\ldots$ can be written as a fraction.
When reduced to lowest terms the sum of the numerator and denominator of this fraction is: | {"ground_truth": "349"} | {} |
deepscaler_math_58 | verifiable_math | For all non-zero numbers $x$ and $y$ such that $x = 1/y$, $\left(x-\frac{1}{x}\right)\left(y+\frac{1}{y}\right)$ equals | {"ground_truth": "x^2-y^2"} | {} |
deepscaler_math_59 | verifiable_math | The values of $k$ for which the equation $2x^2-kx+x+8=0$ will have real and equal roots are: | {"ground_truth": "9 and -7"} | {} |
deepscaler_math_60 | verifiable_math | How many perfect cubes lie between $2^8+1$ and $2^{18}+1$, inclusive? | {"ground_truth": "58"} | {} |
deepscaler_math_61 | verifiable_math | A line that passes through the origin intersects both the line $x = 1$ and the line $y=1+ \frac{\sqrt{3}}{3} x$. The three lines create an equilateral triangle. What is the perimeter of the triangle? | {"ground_truth": "3 + 2\\sqrt{3}"} | {} |
deepscaler_math_62 | verifiable_math | Alicia earns 20 dollars per hour, of which $1.45\%$ is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes? | {"ground_truth": "29"} | {} |
deepscaler_math_63 | verifiable_math | How many positive factors of 36 are also multiples of 4? | {"ground_truth": "3"} | {} |
deepscaler_math_64 | verifiable_math | The numbers $-2, 4, 6, 9$ and $12$ are rearranged according to these rules:
1. The largest isn't first, but it is in one of the first three places.
2. The smallest isn't last, but it is in one of the last three places.
3. The median isn't first or last.
What is the average of the first and last numbers? | {"ground_truth": "6.5"} | {} |
deepscaler_math_65 | verifiable_math | The circumference of the circle with center $O$ is divided into $12$ equal arcs, marked the letters $A$ through $L$ as seen below. What is the number of degrees in the sum of the angles $x$ and $y$? | {"ground_truth": "90"} | {} |
deepscaler_math_66 | verifiable_math | Mary thought of a positive two-digit number. She multiplied it by $3$ and added $11$. Then she switched the digits of the result, obtaining a number between $71$ and $75$, inclusive. What was Mary's number? | {"ground_truth": "12"} | {} |
deepscaler_math_67 | verifiable_math | Trapezoid $ABCD$ has $\overline{AB} \parallel \overline{CD}, BC=CD=43$, and $\overline{AD} \perp \overline{BD}$. Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$, and let $P$ be the midpoint of $\overline{BD}$. Given that $OP=11$, the length of $AD$ can be written in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. What is $m+n$? | {"ground_truth": "194"} | {} |
deepscaler_math_68 | verifiable_math | If $q_1(x)$ and $r_1$ are the quotient and remainder, respectively, when the polynomial $x^8$ is divided by $x + \frac{1}{2}$, and if $q_2(x)$ and $r_2$ are the quotient and remainder, respectively, when $q_1(x)$ is divided by $x + \frac{1}{2}$, then $r_2$ equals | {"ground_truth": "-\\frac{1}{16}"} | {} |
deepscaler_math_69 | verifiable_math | The expression $\frac{1^{4y-1}}{5^{-1}+3^{-1}}$ is equal to: | {"ground_truth": "\\frac{15}{8}"} | {} |
deepscaler_math_70 | verifiable_math | Carrie has a rectangular garden that measures $6$ feet by $8$ feet. She plants the entire garden with strawberry plants. Carrie is able to plant $4$ strawberry plants per square foot, and she harvests an average of $10$ strawberries per plant. How many strawberries can she expect to harvest? | {"ground_truth": "1920"} | {} |
deepscaler_math_71 | verifiable_math | The sides of a triangle have lengths $6.5$, $10$, and $s$, where $s$ is a whole number. What is the smallest possible value of $s$? | {"ground_truth": "4"} | {} |
deepscaler_math_72 | verifiable_math | In the cube $ABCDEFGH$ with opposite vertices $C$ and $E,$ $J$ and $I$ are the midpoints of segments $\overline{FB}$ and $\overline{HD},$ respectively. Let $R$ be the ratio of the area of the cross-section $EJCI$ to the area of one of the faces of the cube. What is $R^2?$ | {"ground_truth": "\\frac{9}{4}"} | {} |
deepscaler_math_73 | verifiable_math | A square and an equilateral triangle have the same perimeter. Let $A$ be the area of the circle circumscribed about the square and $B$ the area of the circle circumscribed around the triangle. Find $A/B$. | {"ground_truth": "\\frac{27}{32}"} | {} |
deepscaler_math_74 | verifiable_math | Quadrilateral $ABCD$ has $AB = BC = CD$, $m\angle ABC = 70^\circ$ and $m\angle BCD = 170^\circ$. What is the degree measure of $\angle BAD$? | {"ground_truth": "85"} | {} |
deepscaler_math_75 | verifiable_math | Let $(a_n)$ and $(b_n)$ be the sequences of real numbers such that
\[ (2 + i)^n = a_n + b_ni \]for all integers $n\geq 0$, where $i = \sqrt{-1}$. What is
\[\sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,?\] | {"ground_truth": "\\frac{7}{16}"} | {} |
deepscaler_math_76 | verifiable_math | An $11 \times 11 \times 11$ wooden cube is formed by gluing together $11^3$ unit cubes. What is the greatest number of unit cubes that can be seen from a single point? | {"ground_truth": "331"} | {} |
deepscaler_math_77 | verifiable_math | If each of two intersecting lines intersects a hyperbola and neither line is tangent to the hyperbola, then the possible number of points of intersection with the hyperbola is: | {"ground_truth": "2, 3, or 4"} | {} |
deepscaler_math_78 | verifiable_math | The Fort Worth Zoo has a number of two-legged birds and a number of four-legged mammals. On one visit to the zoo, Margie counted 200 heads and 522 legs. How many of the animals that Margie counted were two-legged birds? | {"ground_truth": "139"} | {} |
deepscaler_math_79 | verifiable_math | If $\frac{1}{x} - \frac{1}{y} = \frac{1}{z}$, then $z$ equals: | {"ground_truth": "\\frac{xy}{y - x}"} | {} |
deepscaler_math_80 | verifiable_math | A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers with $b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width $1$ foot around the painted rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair $(a,b)$? | {"ground_truth": "2"} | {} |
deepscaler_math_81 | verifiable_math | Ashley, Betty, Carlos, Dick, and Elgin went shopping. Each had a whole number of dollars to spend, and together they had $56$ dollars. The absolute difference between the amounts Ashley and Betty had to spend was $19$ dollars. The absolute difference between the amounts Betty and Carlos had was $7$ dollars, between Carlos and Dick was $5$ dollars, between Dick and Elgin was $4$ dollars, and between Elgin and Ashley was $11$ dollars. How many dollars did Elgin have? | {"ground_truth": "10"} | {} |
deepscaler_math_82 | verifiable_math | A palindrome between $1000$ and $10000$ is chosen at random. What is the probability that it is divisible by $7$? | {"ground_truth": "\\frac{1}{5}"} | {} |
deepscaler_math_83 | verifiable_math | What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is not a divisor of the product of the first $n$ positive integers? | {"ground_truth": "996"} | {} |
deepscaler_math_84 | verifiable_math | In $\triangle ABC$ with right angle at $C$, altitude $CH$ and median $CM$ trisect the right angle. If the area of $\triangle CHM$ is $K$, then the area of $\triangle ABC$ is | {"ground_truth": "4K"} | {} |
deepscaler_math_85 | verifiable_math | Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle? | {"ground_truth": "143"} | {} |
deepscaler_math_86 | verifiable_math | Zara has a collection of $4$ marbles: an Aggie, a Bumblebee, a Steelie, and a Tiger. She wants to display them in a row on a shelf, but does not want to put the Steelie and the Tiger next to one another. In how many ways can she do this? | {"ground_truth": "12"} | {} |
deepscaler_math_87 | verifiable_math | Triangle $ABC$ has vertices $A = (3,0)$, $B = (0,3)$, and $C$, where $C$ is on the line $x + y = 7$. What is the area of $\triangle ABC$? | {"ground_truth": "6"} | {} |
deepscaler_math_88 | verifiable_math | Everyday at school, Jo climbs a flight of $6$ stairs. Jo can take the stairs $1$, $2$, or $3$ at a time. For example, Jo could climb $3$, then $1$, then $2$. In how many ways can Jo climb the stairs? | {"ground_truth": "24"} | {} |
deepscaler_math_89 | verifiable_math | A team won $40$ of its first $50$ games. How many of the remaining $40$ games must this team win so it will have won exactly $70 \%$ of its games for the season? | {"ground_truth": "23"} | {} |
deepscaler_math_90 | verifiable_math | In this diagram, not drawn to scale, Figures $I$ and $III$ are equilateral triangular regions with respective areas of $32\sqrt{3}$ and $8\sqrt{3}$ square inches. Figure $II$ is a square region with area $32$ square inches. Let the length of segment $AD$ be decreased by $12\frac{1}{2}$% of itself, while the lengths of $AB$ and $CD$ remain unchanged. The percent decrease in the area of the square is: | {"ground_truth": "25"} | {} |
deepscaler_math_91 | verifiable_math | Positive integers $a$ and $b$ are each less than $6$. What is the smallest possible value for $2 \cdot a - a \cdot b$? | {"ground_truth": "-15"} | {} |
deepscaler_math_92 | verifiable_math | In square $ABCD$, points $P$ and $Q$ lie on $\overline{AD}$ and $\overline{AB}$, respectively. Segments $\overline{BP}$ and $\overline{CQ}$ intersect at right angles at $R$, with $BR = 6$ and $PR = 7$. What is the area of the square? | {"ground_truth": "117"} | {} |
deepscaler_math_93 | verifiable_math | The equations of $L_1$ and $L_2$ are $y=mx$ and $y=nx$, respectively. Suppose $L_1$ makes twice as large of an angle with the horizontal (measured counterclockwise from the positive x-axis ) as does $L_2$, and that $L_1$ has 4 times the slope of $L_2$. If $L_1$ is not horizontal, then $mn$ is | {"ground_truth": "2"} | {} |
deepscaler_math_94 | verifiable_math | An amusement park has a collection of scale models, with a ratio of $1: 20$, of buildings and other sights from around the country. The height of the United States Capitol is $289$ feet. What is the height in feet of its duplicate to the nearest whole number? | {"ground_truth": "14"} | {} |
deepscaler_math_95 | verifiable_math | Sides $\overline{AB}$ and $\overline{AC}$ of equilateral triangle $ABC$ are tangent to a circle at points $B$ and $C$ respectively. What fraction of the area of $\triangle ABC$ lies outside the circle? | {"ground_truth": "\\frac{4}{3}-\\frac{4\\sqrt{3}\\pi}{27}"} | {} |
deepscaler_math_96 | verifiable_math | Given $0 \le x_0 < 1$, let
\[x_n = \begin{cases} 2x_{n-1} & \text{ if } 2x_{n-1} < 1 \\ 2x_{n-1} - 1 & \text{ if } 2x_{n-1} \ge 1 \end{cases}\]for all integers $n > 0$. For how many $x_0$ is it true that $x_0 = x_5$? | {"ground_truth": "31"} | {} |
deepscaler_math_97 | verifiable_math | Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds 0.1 liters. How tall, in meters, should Logan make his tower? | {"ground_truth": "0.4"} | {} |
deepscaler_math_98 | verifiable_math | In $\triangle PQR$, $PR=15$, $QR=20$, and $PQ=25$. Points $A$ and $B$ lie on $\overline{PQ}$, points $C$ and $D$ lie on $\overline{QR}$, and points $E$ and $F$ lie on $\overline{PR}$, with $PA=QB=QC=RD=RE=PF=5$. Find the area of hexagon $ABCDEF$. | {"ground_truth": "150"} | {} |
deepscaler_math_99 | verifiable_math | Adams plans a profit of $10$ % on the selling price of an article and his expenses are $15$ % of sales. The rate of markup on an article that sells for $ $5.00$ is: | {"ground_truth": "33\\frac {1}{3}\\%"} | {} |
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