data_source stringclasses 6
values | prompt listlengths 1 1 | reward_model dict | question stringlengths 11 5.15k | solution stringlengths 0 454 | answer stringlengths 0 4.59k |
|---|---|---|---|---|---|
math_dapo | [
{
"content": "The trisectors of angles $B$ and $C$ of scalene triangle $ABC$ meet at points $P$ and $Q$ as shown. Angle $A$ measures 39 degrees and angle $QBP$ measures 14 degrees. What is the measure of angle $BPC$? [asy]\nimport olympiad; import geometry; size(150); defaultpen(linewidth(0.8));\ndraw((0,0)--(3... | {
"ground_truth": "133",
"style": "rule-lighteval/MATH_v2"
} | The trisectors of angles $B$ and $C$ of scalene triangle $ABC$ meet at points $P$ and $Q$ as shown. Angle $A$ measures 39 degrees and angle $QBP$ measures 14 degrees. What is the measure of angle $BPC$? [asy]
import olympiad; import geometry; size(150); defaultpen(linewidth(0.8));
draw((0,0)--(3,0)--(4,5)--(0,0)--(2,1.... | 133 | 133 |
deepscaler | [
{
"content": "On the International Space Station, there was an electronic clock displaying time in the HH:MM format. Due to an electromagnetic storm, the device started malfunctioning, causing each digit on the display to either increase by 1 or decrease by 1. What was the actual time when the storm occurred, i... | {
"ground_truth": "18:18",
"style": "rule"
} | On the International Space Station, there was an electronic clock displaying time in the HH:MM format. Due to an electromagnetic storm, the device started malfunctioning, causing each digit on the display to either increase by 1 or decrease by 1. What was the actual time when the storm occurred, if immediately after th... | 18:18 | |
omni_math | [
{
"content": "Let $A B C D$ be a convex trapezoid such that $\\angle B A D=\\angle A D C=90^{\\circ}, A B=20, A D=21$, and $C D=28$. Point $P \\neq A$ is chosen on segment $A C$ such that $\\angle B P D=90^{\\circ}$. Compute $A P$.",
"role": "user"
}
] | {
"ground_truth": "\\frac{143}{5}",
"style": "rule"
} | Let $A B C D$ be a convex trapezoid such that $\angle B A D=\angle A D C=90^{\circ}, A B=20, A D=21$, and $C D=28$. Point $P \neq A$ is chosen on segment $A C$ such that $\angle B P D=90^{\circ}$. Compute $A P$. | \frac{143}{5} | \frac{143}{5} |
math_dapo | [
{
"content": "Let \\( C = \\{ (i,j) | i,j \\text{ integers such that } 0 \\leq i,j \\leq 24 \\} \\). How many squares can be formed in the plane such that all of their vertices are in \\( C \\) and their sides are parallel to the \\( X \\)-axis and \\( Y \\)-axis?",
"role": "user"
}
] | {
"ground_truth": "4900",
"style": "rule-lighteval/MATH_v2"
} | Let \( C = \{ (i,j) | i,j \text{ integers such that } 0 \leq i,j \leq 24 \} \). How many squares can be formed in the plane such that all of their vertices are in \( C \) and their sides are parallel to the \( X \)-axis and \( Y \)-axis? | 4900 | 4900 |
deepscaler | [
{
"content": "Given the positive integer \\( A = \\overline{a_{n} a_{n-1} \\cdots a_{1} a_{0}} \\), where \\( a_{n}, a_{n-1}, \\ldots, a_{0} \\) are all non-zero and not all equal (with \\( n \\) being a positive integer), consider the following cyclic permutations of \\( A \\):\n$$\n\\begin{array}{l}\nA_{1}=\\... | {
"ground_truth": "142857",
"style": "rule"
} | Given the positive integer \( A = \overline{a_{n} a_{n-1} \cdots a_{1} a_{0}} \), where \( a_{n}, a_{n-1}, \ldots, a_{0} \) are all non-zero and not all equal (with \( n \) being a positive integer), consider the following cyclic permutations of \( A \):
$$
\begin{array}{l}
A_{1}=\overline{a_{n-1} \cdots a_{1} a_{0} a_... | 142857 | |
deepscaler | [
{
"content": "A spinner has eight congruent sections, each labeled with numbers 1 to 8. Jane and her brother each spin this spinner once. Jane wins if the non-negative difference of their numbers is less than three; otherwise, her brother wins. Determine the probability of Jane winning. Express your answer as a... | {
"ground_truth": "\\frac{17}{32}",
"style": "rule"
} | A spinner has eight congruent sections, each labeled with numbers 1 to 8. Jane and her brother each spin this spinner once. Jane wins if the non-negative difference of their numbers is less than three; otherwise, her brother wins. Determine the probability of Jane winning. Express your answer as a common fraction. | \frac{17}{32} | |
deepscaler | [
{
"content": "What is the value of the following expression: $\\frac{1}{3}-\\frac{1}{9}+\\frac{1}{27}-\\frac{1}{81}+\\frac{1}{243}$? Express your answer as a common fraction.",
"role": "user"
}
] | {
"ground_truth": "\\frac{61}{243}",
"style": "rule"
} | What is the value of the following expression: $\frac{1}{3}-\frac{1}{9}+\frac{1}{27}-\frac{1}{81}+\frac{1}{243}$? Express your answer as a common fraction. | \frac{61}{243} | |
math_dapo | [
{
"content": "If \\(x \\ge 0\\), then find the value of \\(\\sqrt{x \\sqrt{x \\sqrt{x}}}\\). The original answer is in the format \\(\\sqrt[m]{x^{k}}\\). Please give the value of \\(k + m\\).",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule-lighteval/MATH_v2"
} | If \(x \ge 0\), then find the value of \(\sqrt{x \sqrt{x \sqrt{x}}}\). The original answer is in the format \(\sqrt[m]{x^{k}}\). Please give the value of \(k + m\). | 15 | 15 |
deepscaler | [
{
"content": "In the plane Cartesian coordinate system, the area of the region corresponding to the set of points $\\{(x, y) \\mid(|x|+|3 y|-6)(|3 x|+|y|-6) \\leq 0\\}$ is ________.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | In the plane Cartesian coordinate system, the area of the region corresponding to the set of points $\{(x, y) \mid(|x|+|3 y|-6)(|3 x|+|y|-6) \leq 0\}$ is ________. | 24 | |
deepscaler | [
{
"content": "When Scott completes the square on the quadratic $x^2 + 8x - 1 = 0$, he obtains an equation of the form $(x + a)^2 = b$. What is $b$?",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | When Scott completes the square on the quadratic $x^2 + 8x - 1 = 0$, he obtains an equation of the form $(x + a)^2 = b$. What is $b$? | 17 | |
deepscaler | [
{
"content": "Let $f(x) = |x+1| - |x-4|$.\n(1) Find the range of real number $m$ such that $f(x) \\leq -m^2 + 6m$ always holds;\n(2) Let $m_0$ be the maximum value of $m$. Suppose $a$, $b$, and $c$ are all positive real numbers and $3a + 4b + 5c = m_0$. Find the minimum value of $a^2 + b^2 + c^2$.",
"role":... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | Let $f(x) = |x+1| - |x-4|$.
(1) Find the range of real number $m$ such that $f(x) \leq -m^2 + 6m$ always holds;
(2) Let $m_0$ be the maximum value of $m$. Suppose $a$, $b$, and $c$ are all positive real numbers and $3a + 4b + 5c = m_0$. Find the minimum value of $a^2 + b^2 + c^2$. | \frac{1}{2} | |
deepscaler | [
{
"content": "Consider a parallelogram where each vertex has integer coordinates and is located at $(0,0)$, $(4,5)$, $(11,5)$, and $(7,0)$. Calculate the sum of the perimeter and the area of this parallelogram.",
"role": "user"
}
] | {
"ground_truth": "9\\sqrt{41}",
"style": "rule"
} | Consider a parallelogram where each vertex has integer coordinates and is located at $(0,0)$, $(4,5)$, $(11,5)$, and $(7,0)$. Calculate the sum of the perimeter and the area of this parallelogram. | 9\sqrt{41} | |
deepscaler | [
{
"content": "Let $ABCD$ be a convex quadrilateral with $\\angle ABD = \\angle BCD$ , $AD = 1000$ , $BD = 2000$ , $BC = 2001$ , and $DC = 1999$ . Point $E$ is chosen on segment $DB$ such that $\\angle ABD = \\angle ECD$ . Find $AE$ .",
"role": "user"
}
] | {
"ground_truth": "1000",
"style": "rule"
} | Let $ABCD$ be a convex quadrilateral with $\angle ABD = \angle BCD$ , $AD = 1000$ , $BD = 2000$ , $BC = 2001$ , and $DC = 1999$ . Point $E$ is chosen on segment $DB$ such that $\angle ABD = \angle ECD$ . Find $AE$ . | 1000 | |
math | [
{
"content": "If $a \\div b = 2$ and $b \\div c = \\frac{3}{4}$, what is the value of $c \\div a$? Express your answer as a common fraction.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | If $a \div b = 2$ and $b \div c = \frac{3}{4}$, what is the value of $c \div a$? Express your answer as a common fraction. | \frac{2}{3} | \frac{2}{3} |
deepscaler | [
{
"content": "Let $S$ be the set of positive real numbers. Let $f : S \\to \\mathbb{R}$ be a function such that\n\\[f(x) f(y) = f(xy) + 2023 \\left( \\frac{2}{x} + \\frac{2}{y} + 2022 \\right)\\] for all $x, y > 0.$\n\nLet $n$ be the number of possible values of $f(2)$, and let $s$ be the sum of all possible va... | {
"ground_truth": "2023",
"style": "rule"
} | Let $S$ be the set of positive real numbers. Let $f : S \to \mathbb{R}$ be a function such that
\[f(x) f(y) = f(xy) + 2023 \left( \frac{2}{x} + \frac{2}{y} + 2022 \right)\] for all $x, y > 0.$
Let $n$ be the number of possible values of $f(2)$, and let $s$ be the sum of all possible values of $f(2)$. Find $n \times s.... | 2023 | |
math | [
{
"content": "Find the largest value of $x$ that satisfies the equation $|x-5|=12$.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | Find the largest value of $x$ that satisfies the equation $|x-5|=12$. | 17 | 17 |
math_dapo | [
{
"content": "If $10^{51} - 9$ is written as an integer in standard form, what is the sum of the integer's digits?",
"role": "user"
}
] | {
"ground_truth": "451",
"style": "rule-lighteval/MATH_v2"
} | If $10^{51} - 9$ is written as an integer in standard form, what is the sum of the integer's digits? | 451 | 451 |
deepscaler | [
{
"content": "Given a $24$-inch by $30$-inch pan of brownies, cut into pieces that measure $3$ inches by $4$ inches. Calculate the number of pieces of brownie the pan contains.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | Given a $24$-inch by $30$-inch pan of brownies, cut into pieces that measure $3$ inches by $4$ inches. Calculate the number of pieces of brownie the pan contains. | 60 | |
deepscaler | [
{
"content": "Given that a high school senior year has 12 classes, with exactly 8 classes to be proctored by their own homeroom teachers, find the number of different proctoring arrangements for the math exam.",
"role": "user"
}
] | {
"ground_truth": "4455",
"style": "rule"
} | Given that a high school senior year has 12 classes, with exactly 8 classes to be proctored by their own homeroom teachers, find the number of different proctoring arrangements for the math exam. | 4455 | |
deepscaler | [
{
"content": "A chord PQ of the left branch of the hyperbola $x^2 - y^2 = 4$ passes through its left focus $F_1$, and the length of $|PQ|$ is 7. If $F_2$ is the right focus of the hyperbola, then the perimeter of $\\triangle PF_2Q$ is.",
"role": "user"
}
] | {
"ground_truth": "22",
"style": "rule"
} | A chord PQ of the left branch of the hyperbola $x^2 - y^2 = 4$ passes through its left focus $F_1$, and the length of $|PQ|$ is 7. If $F_2$ is the right focus of the hyperbola, then the perimeter of $\triangle PF_2Q$ is. | 22 | |
deepscaler | [
{
"content": "How many parallelograms with sides 1 and 2, and angles $60^{\\circ}$ and $120^{\\circ}$ can be placed at most inside a regular hexagon with side length 3?",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | How many parallelograms with sides 1 and 2, and angles $60^{\circ}$ and $120^{\circ}$ can be placed at most inside a regular hexagon with side length 3? | 12 | |
deepscaler | [
{
"content": "$\\frac{2+4+6+\\cdots + 34}{3+6+9+\\cdots+51}=$",
"role": "user"
}
] | {
"ground_truth": "$\\frac{2}{3}$",
"style": "rule"
} | $\frac{2+4+6+\cdots + 34}{3+6+9+\cdots+51}=$ | $\frac{2}{3}$ | 1. **Identify the sequences and their properties:**
- The numerator is an arithmetic sequence with the first term $a = 2$, common difference $d = 2$, and last term $l = 34$.
- The denominator is an arithmetic sequence with the first term $a = 3$, common difference $d = 3$, and last term $l = 51$.
2. **Determine ... |
math | [
{
"content": "It is now 12:00:00 midnight, as read on a 12-hour digital clock. In 122 hours, 39 minutes and 44 seconds the time will be $A:B:C$. What is the value of $A + B + C$?",
"role": "user"
}
] | {
"ground_truth": "85",
"style": "rule"
} | It is now 12:00:00 midnight, as read on a 12-hour digital clock. In 122 hours, 39 minutes and 44 seconds the time will be $A:B:C$. What is the value of $A + B + C$? | 85 | 85 |
math_dapo | [
{
"content": "A right circular cone is sliced into four pieces by planes parallel to its base, as shown in the figure. All of these pieces have the same height. What is the ratio of the volume of the second-largest piece to the volume of the largest piece? Express your answer as a common fraction.\n[asy]\nsize(... | {
"ground_truth": "56",
"style": "rule-lighteval/MATH_v2"
} | A right circular cone is sliced into four pieces by planes parallel to its base, as shown in the figure. All of these pieces have the same height. What is the ratio of the volume of the second-largest piece to the volume of the largest piece? Express your answer as a common fraction.
[asy]
size(150);
pair A, B, C, D, E... | 56 | 56 |
deepscaler | [
{
"content": "The roots of the equation $ax^2 + bx + c = 0$ will be reciprocal if:",
"role": "user"
}
] | {
"ground_truth": "c = a",
"style": "rule"
} | The roots of the equation $ax^2 + bx + c = 0$ will be reciprocal if: | c = a | 1. **Start by normalizing the quadratic equation**: Given the quadratic equation $ax^2 + bx + c = 0$, we can divide each term by $a$ (assuming $a \neq 0$) to simplify the equation:
\[
x^2 + \frac{b}{a}x + \frac{c}{a} = 0
\]
2. **Identify the roots**: Let the roots of the equation be $r$ and $s$. According to ... |
deepscaler | [
{
"content": "Find how many integer values of \\( x \\) are there such that \\( \\lceil{\\sqrt{x}}\\rceil=18 \\)?",
"role": "user"
}
] | {
"ground_truth": "35",
"style": "rule"
} | Find how many integer values of \( x \) are there such that \( \lceil{\sqrt{x}}\rceil=18 \)? | 35 | |
deepscaler | [
{
"content": "During a secret meeting, 20 trainees elect their favorite supervisor. Each trainee votes for two supervisors. It is known that for any two trainees, there is always at least one supervisor for whom both have voted. What is the minimum number of votes received by the supervisor who wins the electio... | {
"ground_truth": "14",
"style": "rule"
} | During a secret meeting, 20 trainees elect their favorite supervisor. Each trainee votes for two supervisors. It is known that for any two trainees, there is always at least one supervisor for whom both have voted. What is the minimum number of votes received by the supervisor who wins the election? | 14 | |
math_dapo | [
{
"content": "Two \\(4 \\times 4\\) squares intersect at right angles, bisecting their intersecting sides, as shown. The circle's diameter is the segment between the two points of intersection. Find the area of the shaded region created by removing the circle from the squares. The answer is in the form \\(k - m... | {
"ground_truth": "30",
"style": "rule-lighteval/MATH_v2"
} | Two \(4 \times 4\) squares intersect at right angles, bisecting their intersecting sides, as shown. The circle's diameter is the segment between the two points of intersection. Find the area of the shaded region created by removing the circle from the squares. The answer is in the form \(k - m\pi\), please provide the ... | 30 | 30 |
deepscaler | [
{
"content": "A teacher has a class with $24$ students in it. If she wants to split the students into equal groups of at most $10$ students each, what is the least number of groups that she needs?",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | A teacher has a class with $24$ students in it. If she wants to split the students into equal groups of at most $10$ students each, what is the least number of groups that she needs? | 3 | |
deepscaler | [
{
"content": "Given a geometric sequence $\\{a_n\\}$ with the first term $\\frac{3}{2}$ and common ratio $- \\frac{1}{2}$, the sum of the first $n$ terms is $S_n$. If for any $n \\in N^*$, it holds that $S_n - \\frac{1}{S_n} \\in [s, t]$, then the minimum value of $t-s$ is \\_\\_\\_\\_\\_\\_.",
"role": "use... | {
"ground_truth": "\\frac{17}{12}",
"style": "rule"
} | Given a geometric sequence $\{a_n\}$ with the first term $\frac{3}{2}$ and common ratio $- \frac{1}{2}$, the sum of the first $n$ terms is $S_n$. If for any $n \in N^*$, it holds that $S_n - \frac{1}{S_n} \in [s, t]$, then the minimum value of $t-s$ is \_\_\_\_\_\_. | \frac{17}{12} | |
deepscaler | [
{
"content": "Given point $P(1, 2, 3)$, the symmetric point of $P$ about the $y$-axis is $P_1$, and the symmetric point of $P$ about the coordinate plane $xOz$ is $P_2$. Find the distance $|P_1P_2|$.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{14}",
"style": "rule"
} | Given point $P(1, 2, 3)$, the symmetric point of $P$ about the $y$-axis is $P_1$, and the symmetric point of $P$ about the coordinate plane $xOz$ is $P_2$. Find the distance $|P_1P_2|$. | 2\sqrt{14} | |
deepscaler | [
{
"content": "Two rays with common endpoint $O$ form a $30^\\circ$ angle. Point $A$ lies on one ray, point $B$ on the other ray, and $AB=1$. What is the maximum possible length of $\\overline{OB}$?",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | Two rays with common endpoint $O$ form a $30^\circ$ angle. Point $A$ lies on one ray, point $B$ on the other ray, and $AB=1$. What is the maximum possible length of $\overline{OB}$? | 2 | |
math_dapo | [
{
"content": "Points $B$, $D$, and $J$ are midpoints of the sides of right triangle $ACG$. Points $K$, $E$, $I$ are midpoints of the sides of triangle $JDG$, etc. If the dividing and shading process is done 100 times (the first three are shown) and $AC=CG=6$, then find the total area of the shaded triangles rou... | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | Points $B$, $D$, and $J$ are midpoints of the sides of right triangle $ACG$. Points $K$, $E$, $I$ are midpoints of the sides of triangle $JDG$, etc. If the dividing and shading process is done 100 times (the first three are shown) and $AC=CG=6$, then find the total area of the shaded triangles rounded to the nearest in... | 6 | 6 |
deepscaler | [
{
"content": "How many perfect squares are two-digit and divisible by $3?$",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | How many perfect squares are two-digit and divisible by $3?$ | 2 | |
math | [
{
"content": "Let $a$ and $b$ be the roots of $k(x^2 - x) + x + 5 = 0.$ Let $k_1$ and $k_2$ be the values of $k$ for which $a$ and $b$ satisfy\n\\[\\frac{a}{b} + \\frac{b}{a} = \\frac{4}{5}.\\]Find\n\\[\\frac{k_1}{k_2} + \\frac{k_2}{k_1}.\\]",
"role": "user"
}
] | {
"ground_truth": "254",
"style": "rule"
} | Let $a$ and $b$ be the roots of $k(x^2 - x) + x + 5 = 0.$ Let $k_1$ and $k_2$ be the values of $k$ for which $a$ and $b$ satisfy
\[\frac{a}{b} + \frac{b}{a} = \frac{4}{5}.\]Find
\[\frac{k_1}{k_2} + \frac{k_2}{k_1}.\] | 254 | 254 |
deepscaler | [
{
"content": "Masha and the Bear ate a basket of raspberries and 60 pies, starting and finishing at the same time. Initially, Masha ate raspberries while the Bear ate pies, and then they switched at some point. The Bear ate raspberries 6 times faster than Masha and pies 3 times faster. How many pies did the Bea... | {
"ground_truth": "54",
"style": "rule"
} | Masha and the Bear ate a basket of raspberries and 60 pies, starting and finishing at the same time. Initially, Masha ate raspberries while the Bear ate pies, and then they switched at some point. The Bear ate raspberries 6 times faster than Masha and pies 3 times faster. How many pies did the Bear eat if the Bear ate ... | 54 | |
deepscaler | [
{
"content": "A construction team has 6 projects (A, B, C, D, E, F) that need to be completed separately. Project B must start after project A is completed, project C must start after project B is completed, and project D must immediately follow project C. Determine the number of different ways to schedule thes... | {
"ground_truth": "20",
"style": "rule"
} | A construction team has 6 projects (A, B, C, D, E, F) that need to be completed separately. Project B must start after project A is completed, project C must start after project B is completed, and project D must immediately follow project C. Determine the number of different ways to schedule these 6 projects. | 20 | |
math | [
{
"content": "Let $n$ be a positive integer and $a,b$ be invertible integers modulo $n$ such that $a\\equiv b^{-1}\\pmod n$. What is the remainder when $ab$ is divided by $n$?",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | Let $n$ be a positive integer and $a,b$ be invertible integers modulo $n$ such that $a\equiv b^{-1}\pmod n$. What is the remainder when $ab$ is divided by $n$? | 1 | 1 |
math_dapo | [
{
"content": "Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get?",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule-lighteval/MATH_v2"
} | Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get? | 32 | 32 |
deepscaler | [
{
"content": "An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point?",
"rol... | {
"ground_truth": "221",
"style": "rule"
} | An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point? | 221 | |
deepscaler | [
{
"content": "Two fair 6-sided dice are rolled. What is the probability at least one of the dice shows a 1?",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{11}{36}",
"style": "rule"
} | Two fair 6-sided dice are rolled. What is the probability at least one of the dice shows a 1? | \dfrac{11}{36} | |
math_dapo | [
{
"content": "For how many ordered triples $(x,y,z)$ of nonnegative integers less than $20$ are there exactly two distinct elements in the set $\\{i^x, (1+i)^y, z\\}$, where $i^2 = -1$?",
"role": "user"
}
] | {
"ground_truth": "225",
"style": "rule-lighteval/MATH_v2"
} | For how many ordered triples $(x,y,z)$ of nonnegative integers less than $20$ are there exactly two distinct elements in the set $\{i^x, (1+i)^y, z\}$, where $i^2 = -1$? | 225 | 225 |
deepscaler | [
{
"content": "From the set $\\{1, 2, 3, 4, \\ldots, 20\\}$, select four different numbers $a, b, c, d$ such that $a+c=b+d$. If the order of $a, b, c, d$ does not matter, calculate the total number of ways to select these numbers.",
"role": "user"
}
] | {
"ground_truth": "525",
"style": "rule"
} | From the set $\{1, 2, 3, 4, \ldots, 20\}$, select four different numbers $a, b, c, d$ such that $a+c=b+d$. If the order of $a, b, c, d$ does not matter, calculate the total number of ways to select these numbers. | 525 | |
deepscaler | [
{
"content": "The area of a square equals the square of a length of the side of the square. The perimeter of a square equals the sum of the lengths of all four sides. The sum of the areas of two squares is 65, while the difference in their areas is 33. Find the sum of their perimeters.",
"role": "user"
}... | {
"ground_truth": "44",
"style": "rule"
} | The area of a square equals the square of a length of the side of the square. The perimeter of a square equals the sum of the lengths of all four sides. The sum of the areas of two squares is 65, while the difference in their areas is 33. Find the sum of their perimeters. | 44 | |
deepscaler | [
{
"content": "The length of the longer side of rectangle \\(R'\\) is \\(15\\%\\) more than the length of a side of square \\(S'\\). The length of the shorter side of rectangle \\(R'\\) is \\(5\\%\\) less than the length of a side of square \\(S'\\). What is the ratio of the area of rectangle \\(R'\\) to the are... | {
"ground_truth": "\\frac{10925}{10000}",
"style": "rule"
} | The length of the longer side of rectangle \(R'\) is \(15\%\) more than the length of a side of square \(S'\). The length of the shorter side of rectangle \(R'\) is \(5\%\) less than the length of a side of square \(S'\). What is the ratio of the area of rectangle \(R'\) to the area of square \(S'\)? Express your answe... | \frac{10925}{10000} | |
deepscaler | [
{
"content": "A sequence of figures is formed using tiles. Each tile is an equilateral triangle with side length 7 cm. The first figure consists of 1 tile. Each figure after the first is formed by adding 1 tile to the previous figure. How many tiles are used to form the figure in the sequence with perimeter 91 ... | {
"ground_truth": "11",
"style": "rule"
} | A sequence of figures is formed using tiles. Each tile is an equilateral triangle with side length 7 cm. The first figure consists of 1 tile. Each figure after the first is formed by adding 1 tile to the previous figure. How many tiles are used to form the figure in the sequence with perimeter 91 cm? | 11 | The first figure consists of one tile with perimeter $3 \times 7 \mathrm{~cm} = 21 \mathrm{~cm}$. Each time an additional tile is added, the perimeter of the figure increases by 7 cm (one side length of a tile), because one side length of the previous figure is 'covered up' and two new side lengths of a tile are added ... |
deepscaler | [
{
"content": "Given the sequence $\\{a_n\\}$ satisfies $\\{a_1=2, a_2=1,\\}$ and $\\frac{a_n \\cdot a_{n-1}}{a_{n-1}-a_n}=\\frac{a_n \\cdot a_{n+1}}{a_n-a_{n+1}}(n\\geqslant 2)$, determine the $100^{\\text{th}}$ term of the sequence $\\{a_n\\}$.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{50}",
"style": "rule"
} | Given the sequence $\{a_n\}$ satisfies $\{a_1=2, a_2=1,\}$ and $\frac{a_n \cdot a_{n-1}}{a_{n-1}-a_n}=\frac{a_n \cdot a_{n+1}}{a_n-a_{n+1}}(n\geqslant 2)$, determine the $100^{\text{th}}$ term of the sequence $\{a_n\}$. | \frac{1}{50} | |
deepscaler | [
{
"content": "Given the function \\( f:\\{1,2, \\cdots, 10\\} \\rightarrow\\{1,2,3,4,5\\} \\), and for each \\( k=1,2, \\cdots, 9 \\), it is true that \\( |f(k+1)-f(k)| \\geq 3 \\). Find the number of functions \\( f \\) that satisfy these conditions.",
"role": "user"
}
] | {
"ground_truth": "288",
"style": "rule"
} | Given the function \( f:\{1,2, \cdots, 10\} \rightarrow\{1,2,3,4,5\} \), and for each \( k=1,2, \cdots, 9 \), it is true that \( |f(k+1)-f(k)| \geq 3 \). Find the number of functions \( f \) that satisfy these conditions. | 288 | |
deepscaler | [
{
"content": "A positive unknown number less than 2022 was written on the board next to the number 2022. Then, one of the numbers on the board was replaced by their arithmetic mean. This replacement was done 9 more times, and the arithmetic mean was always an integer. Find the smaller of the numbers that were i... | {
"ground_truth": "998",
"style": "rule"
} | A positive unknown number less than 2022 was written on the board next to the number 2022. Then, one of the numbers on the board was replaced by their arithmetic mean. This replacement was done 9 more times, and the arithmetic mean was always an integer. Find the smaller of the numbers that were initially written on th... | 998 | |
deepscaler | [
{
"content": "How many lattice points (points with integer coordinates) are inside (but not on the boundary) the region formed by the right branch of the hyperbola $x^{2} - y^{2} = 1$ and the line $x = 100$?",
"role": "user"
}
] | {
"ground_truth": "9800",
"style": "rule"
} | How many lattice points (points with integer coordinates) are inside (but not on the boundary) the region formed by the right branch of the hyperbola $x^{2} - y^{2} = 1$ and the line $x = 100$? | 9800 | |
deepscaler | [
{
"content": "How many ways are there to put 6 balls in 4 boxes if the balls are not distinguishable but the boxes are?",
"role": "user"
}
] | {
"ground_truth": "84",
"style": "rule"
} | How many ways are there to put 6 balls in 4 boxes if the balls are not distinguishable but the boxes are? | 84 | |
amc | [
{
"content": "In the small country of Mathland, all automobile license plates have four symbols. The first must be a vowel (A, E, I, O, or U), the second and third must be two different letters among the 21 non-vowels, and the fourth must be a digit (0 through 9). If the symbols are chosen at random subject to ... | {
"ground_truth": "\\frac{1}{21,000}",
"style": "rule"
} | In the small country of Mathland, all automobile license plates have four symbols. The first must be a vowel (A, E, I, O, or U), the second and third must be two different letters among the 21 non-vowels, and the fourth must be a digit (0 through 9). If the symbols are chosen at random subject to these conditions, what... | \frac{1}{21,000} | \frac{1}{21,000} |
deepscaler | [
{
"content": "Points $A,B,C$ and $D$ lie on a line in that order, with $AB = CD$ and $BC = 16$. Point $E$ is not on the line, and $BE = CE = 13$. The perimeter of $\\triangle AED$ is three times the perimeter of $\\triangle BEC$. Find $AB$.\nA) $\\frac{32}{3}$\nB) $\\frac{34}{3}$\nC) $\\frac{36}{3}$\nD) $\\frac... | {
"ground_truth": "\\frac{34}{3}",
"style": "rule"
} | Points $A,B,C$ and $D$ lie on a line in that order, with $AB = CD$ and $BC = 16$. Point $E$ is not on the line, and $BE = CE = 13$. The perimeter of $\triangle AED$ is three times the perimeter of $\triangle BEC$. Find $AB$.
A) $\frac{32}{3}$
B) $\frac{34}{3}$
C) $\frac{36}{3}$
D) $\frac{38}{3}$ | \frac{34}{3} | |
math_dapo | [
{
"content": "How many positive integers less than 2015 have exactly 9 positive integer divisors?",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule-lighteval/MATH_v2"
} | How many positive integers less than 2015 have exactly 9 positive integer divisors? | 13 | 13 |
math_dapo | [
{
"content": "Let $ABCD$ be a regular tetrahedron with side length $12$. Select points $E$, $F$, $G$, $H$ on $AC$, $BC$, $BD$, and $AD$, respectively, such that $AE = BF = BG = AH = 3$. Compute the area of quadrilateral $EFGH$.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule-lighteval/MATH_v2"
} | Let $ABCD$ be a regular tetrahedron with side length $12$. Select points $E$, $F$, $G$, $H$ on $AC$, $BC$, $BD$, and $AD$, respectively, such that $AE = BF = BG = AH = 3$. Compute the area of quadrilateral $EFGH$. | 27 | 27 |
deepscaler | [
{
"content": "Marla has a large white cube that has an edge of 10 feet. She also has enough green paint to cover 300 square feet. Marla uses all the paint to create a white square centered on each face, surrounded by a green border. What is the area of one of the white squares, in square feet?",
"role": "us... | {
"ground_truth": "50",
"style": "rule"
} | Marla has a large white cube that has an edge of 10 feet. She also has enough green paint to cover 300 square feet. Marla uses all the paint to create a white square centered on each face, surrounded by a green border. What is the area of one of the white squares, in square feet? | 50 | 1. **Calculate the total surface area of the cube**:
The cube has 6 faces, and each face is a square with an edge length of 10 feet. The area of one face is:
\[
10 \times 10 = 100 \text{ square feet}
\]
Therefore, the total surface area of the cube is:
\[
6 \times 100 = 600 \text{ square feet}
... |
deepscaler | [
{
"content": "Determine how many integer values of $n$ between 1 and 180 inclusive ensure that the decimal representation of $\\frac{n}{180}$ terminates.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | Determine how many integer values of $n$ between 1 and 180 inclusive ensure that the decimal representation of $\frac{n}{180}$ terminates. | 60 | |
deepscaler | [
{
"content": "Let $p>q$ be primes, such that $240 \\nmid p^4-q^4$ . Find the maximal value of $\\frac{q} {p}$ .",
"role": "user"
}
] | {
"ground_truth": "2/3",
"style": "rule"
} | Let $p>q$ be primes, such that $240 \nmid p^4-q^4$ . Find the maximal value of $\frac{q} {p}$ . | 2/3 | |
deepscaler | [
{
"content": "Let \\( a \\) and \\( b \\) be real numbers, and consider the function \\( f(x) = x^{3} + a x^{2} + b x \\). If there exist three real numbers \\( x_{1}, x_{2}, x_{3} \\) such that \\( x_{1} + 1 \\leqslant x_{2} \\leqslant x_{3} - 1 \\), and \\( f(x_{1}) = f(x_{2}) = f(x_{3}) \\), find the minimum... | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | Let \( a \) and \( b \) be real numbers, and consider the function \( f(x) = x^{3} + a x^{2} + b x \). If there exist three real numbers \( x_{1}, x_{2}, x_{3} \) such that \( x_{1} + 1 \leqslant x_{2} \leqslant x_{3} - 1 \), and \( f(x_{1}) = f(x_{2}) = f(x_{3}) \), find the minimum value of \( |a| + 2|b| \). | \sqrt{3} | |
math_dapo | [
{
"content": "The ratio of losses to wins for Kyle's team is 3 to 2. If the team had played the same number of games, but had won twice as many of its games, what would the ratio of losses to wins have been? Express your answer as a common fraction.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Ple... | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | The ratio of losses to wins for Kyle's team is 3 to 2. If the team had played the same number of games, but had won twice as many of its games, what would the ratio of losses to wins have been? Express your answer as a common fraction.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value o... | 5 | 5 |
omni_math | [
{
"content": "Let $P$ be a regular $2006$-gon. A diagonal is called [i]good[/i] if its endpoints divide the boundary of $P$ into two parts, each composed of an odd number of sides of $P$. The sides of $P$ are also called [i]good[/i].\nSuppose $P$ has been dissected into triangles by $2003$ diagonals, no two of ... | {
"ground_truth": "1003",
"style": "rule"
} | Let $P$ be a regular $2006$-gon. A diagonal is called [i]good[/i] if its endpoints divide the boundary of $P$ into two parts, each composed of an odd number of sides of $P$. The sides of $P$ are also called [i]good[/i].
Suppose $P$ has been dissected into triangles by $2003$ diagonals, no two of which have a common poi... | 1003 | 1003 |
math_dapo | [
{
"content": "Compute\n\\[\\sin^2 4^\\circ + \\sin^2 8^\\circ + \\sin^2 12^\\circ + \\dots + \\sin^2 176^\\circ.\\]The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.",
"role": "user"
}
] | {
"ground_truth": "47",
"style": "rule-lighteval/MATH_v2"
} | Compute
\[\sin^2 4^\circ + \sin^2 8^\circ + \sin^2 12^\circ + \dots + \sin^2 176^\circ.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 47 | 47 |
deepscaler | [
{
"content": "Given that $a-b=5$ and $a^2+b^2=35$, find $a^3-b^3$.",
"role": "user"
}
] | {
"ground_truth": "200",
"style": "rule"
} | Given that $a-b=5$ and $a^2+b^2=35$, find $a^3-b^3$. | 200 | |
deepscaler | [
{
"content": "A certain high school encourages students to spontaneously organize various sports competitions. Two students, A and B, play table tennis matches in their spare time. The competition adopts a \"best of seven games\" format (i.e., the first player to win four games wins the match and the competitio... | {
"ground_truth": "\\frac{19}{9}",
"style": "rule"
} | A certain high school encourages students to spontaneously organize various sports competitions. Two students, A and B, play table tennis matches in their spare time. The competition adopts a "best of seven games" format (i.e., the first player to win four games wins the match and the competition ends). Assuming that t... | \frac{19}{9} | |
deepscaler | [
{
"content": "The slope of the tangent line to the curve $y=\\frac{1}{3}{x^3}-\\frac{2}{x}$ at $x=1$ is $\\alpha$. Find $\\frac{{sin\\alpha cos2\\alpha}}{{sin\\alpha+cos\\alpha}}$.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{3}{5}",
"style": "rule"
} | The slope of the tangent line to the curve $y=\frac{1}{3}{x^3}-\frac{2}{x}$ at $x=1$ is $\alpha$. Find $\frac{{sin\alpha cos2\alpha}}{{sin\alpha+cos\alpha}}$. | -\frac{3}{5} | |
deepscaler | [
{
"content": "Ana's monthly salary was $2000$ in May. In June she received a 20% raise. In July she received a 20% pay cut. After the two changes in June and July, Ana's monthly salary was",
"role": "user"
}
] | {
"ground_truth": "1920",
"style": "rule"
} | Ana's monthly salary was $2000$ in May. In June she received a 20% raise. In July she received a 20% pay cut. After the two changes in June and July, Ana's monthly salary was | 1920 | 1. **Calculate the salary after the raise in June:**
Ana's initial salary in May is $2000. A 20\% raise means her salary is increased by $2000 \times 20\% = $400. Therefore, her new salary in June becomes:
\[
2000 + 400 = 2400
\]
Alternatively, this can be calculated directly by multiplying her original ... |
amc | [
{
"content": "Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (list... | {
"ground_truth": "80",
"style": "rule"
} | Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order)... | 80 | 80 |
deepscaler | [
{
"content": "Rewrite $\\sqrt[3]{2^6\\cdot3^3\\cdot11^3}$ as an integer.",
"role": "user"
}
] | {
"ground_truth": "132",
"style": "rule"
} | Rewrite $\sqrt[3]{2^6\cdot3^3\cdot11^3}$ as an integer. | 132 | |
math_dapo | [
{
"content": "Larry always orders pizza with exactly two of his three favorite toppings: pepperoni, bacon, and sausage. If he has ordered a total of $600$ pizzas and has had each topping equally often, how many pizzas has he ordered with pepperoni? Provide your answer as an integer.",
"role": "user"
}
] | {
"ground_truth": "400",
"style": "rule-lighteval/MATH_v2"
} | Larry always orders pizza with exactly two of his three favorite toppings: pepperoni, bacon, and sausage. If he has ordered a total of $600$ pizzas and has had each topping equally often, how many pizzas has he ordered with pepperoni? Provide your answer as an integer. | 400 | 400 |
math_dapo | [
{
"content": "A cross-pentomino is a shape that consists of a unit square and four other unit squares, each sharing a different edge with the first square. If a cross-pentomino is inscribed in a circle of radius $R$, what is $100R^2$?",
"role": "user"
}
] | {
"ground_truth": "250",
"style": "rule-lighteval/MATH_v2"
} | A cross-pentomino is a shape that consists of a unit square and four other unit squares, each sharing a different edge with the first square. If a cross-pentomino is inscribed in a circle of radius $R$, what is $100R^2$? | 250 | 250 |
math_dapo | [
{
"content": "Find all triples $( p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^p-1+y$ and $x+y^p-1$ are both powers of $p.$ Provide the sum of the elements in all triples when k=1.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule-lighteval/MATH_v2"
} | Find all triples $( p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^p-1+y$ and $x+y^p-1$ are both powers of $p.$ Provide the sum of the elements in all triples when k=1. | 24 | 24 |
deepscaler | [
{
"content": "A train has 18 identical cars. In some of the cars, half of the seats are free, in others, one third of the seats are free, and in the remaining cars, all the seats are occupied. In the entire train, exactly one ninth of all seats are free. How many cars have all seats occupied?",
"role": "use... | {
"ground_truth": "13",
"style": "rule"
} | A train has 18 identical cars. In some of the cars, half of the seats are free, in others, one third of the seats are free, and in the remaining cars, all the seats are occupied. In the entire train, exactly one ninth of all seats are free. How many cars have all seats occupied? | 13 | |
math_dapo | [
{
"content": "Let $a, b, c > 1$ be positive real numbers such that $a^{\\log_b c} = 27$, $b^{\\log_c a} = 81$, and $c^{\\log_a b} = 243$. Then the value of $\\log_3{abc}$ can be written as $\\sqrt{x} + \\sqrt{y} + \\sqrt{z}$ for positive integers $x, y,$ and $z$. Find $x + y + z$.",
"role": "user"
}
] | {
"ground_truth": "47",
"style": "rule-lighteval/MATH_v2"
} | Let $a, b, c > 1$ be positive real numbers such that $a^{\log_b c} = 27$, $b^{\log_c a} = 81$, and $c^{\log_a b} = 243$. Then the value of $\log_3{abc}$ can be written as $\sqrt{x} + \sqrt{y} + \sqrt{z}$ for positive integers $x, y,$ and $z$. Find $x + y + z$. | 47 | 47 |
deepscaler | [
{
"content": "A fair coin is tossed 4 times. Calculate the probability of getting at least two consecutive heads.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | A fair coin is tossed 4 times. Calculate the probability of getting at least two consecutive heads. | \frac{1}{2} | |
deepscaler | [
{
"content": "Given Liam has written one integer three times and another integer four times. The sum of these seven numbers is 131, and one of the numbers is 17, determine the value of the other number.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | Given Liam has written one integer three times and another integer four times. The sum of these seven numbers is 131, and one of the numbers is 17, determine the value of the other number. | 21 | |
deepscaler | [
{
"content": "Given the universal set $U=\\{1, 2, 3, 4, 5, 6, 7, 8\\}$, a set $A=\\{a_1, a_2, a_3, a_4\\}$ is formed by selecting any four elements from $U$, and the set of the remaining four elements is denoted as $\\complement_U A=\\{b_1, b_2, b_3, b_4\\}$. If $a_1+a_2+a_3+a_4 < b_1+b_2+b_3+b_4$, then the num... | {
"ground_truth": "31",
"style": "rule"
} | Given the universal set $U=\{1, 2, 3, 4, 5, 6, 7, 8\}$, a set $A=\{a_1, a_2, a_3, a_4\}$ is formed by selecting any four elements from $U$, and the set of the remaining four elements is denoted as $\complement_U A=\{b_1, b_2, b_3, b_4\}$. If $a_1+a_2+a_3+a_4 < b_1+b_2+b_3+b_4$, then the number of ways to form set $A$ i... | 31 | |
deepscaler | [
{
"content": "In the final phase of a professional bowling competition, the top five players compete as follows: first, the fifth and fourth place players compete, and the loser gets the 5th place prize; the winner then competes with the third place player, and the loser gets the 4th place prize; the winner the... | {
"ground_truth": "16",
"style": "rule"
} | In the final phase of a professional bowling competition, the top five players compete as follows: first, the fifth and fourth place players compete, and the loser gets the 5th place prize; the winner then competes with the third place player, and the loser gets the 4th place prize; the winner then competes with the se... | 16 | |
deepscaler | [
{
"content": "A plane flies from city A to city B against a wind in 120 minutes. On the return trip with the wind, it takes 10 minutes less than it would in still air. Determine the time in minutes for the return trip.",
"role": "user"
}
] | {
"ground_truth": "110",
"style": "rule"
} | A plane flies from city A to city B against a wind in 120 minutes. On the return trip with the wind, it takes 10 minutes less than it would in still air. Determine the time in minutes for the return trip. | 110 | |
deepscaler | [
{
"content": "Cara is in a group photo with her seven friends. If Cara must stand between two of her friends, how many different possible pairs of friends could she be standing between?",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | Cara is in a group photo with her seven friends. If Cara must stand between two of her friends, how many different possible pairs of friends could she be standing between? | 21 | |
deepscaler | [
{
"content": "Let \\( n \\) be a positive integer. Given a real number \\( x \\), let \\( \\lfloor x \\rfloor \\) be the greatest integer less than or equal to \\( x \\). For example, \\( \\lfloor 2.4 \\rfloor = 2 \\), \\( \\lfloor 3 \\rfloor = 3 \\), and \\( \\lfloor \\pi \\rfloor = 3 \\). Define a sequence \\... | {
"ground_truth": "126",
"style": "rule"
} | Let \( n \) be a positive integer. Given a real number \( x \), let \( \lfloor x \rfloor \) be the greatest integer less than or equal to \( x \). For example, \( \lfloor 2.4 \rfloor = 2 \), \( \lfloor 3 \rfloor = 3 \), and \( \lfloor \pi \rfloor = 3 \). Define a sequence \( a_1, a_2, a_3, \ldots \) where \( a_1 = n \)... | 126 | |
math_dapo | [
{
"content": "What is the greatest three-digit divisor of $111777$?",
"role": "user"
}
] | {
"ground_truth": "703",
"style": "rule-lighteval/MATH_v2"
} | What is the greatest three-digit divisor of $111777$? | 703 | 703 |
deepscaler | [
{
"content": "Given that Jeff, Maria, and Lee paid $90, $150, and $210 respectively, find j - m where Jeff gave Lee $j dollars and Maria gave Lee $m dollars to settle the debts such that everyone paid equally.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | Given that Jeff, Maria, and Lee paid $90, $150, and $210 respectively, find j - m where Jeff gave Lee $j dollars and Maria gave Lee $m dollars to settle the debts such that everyone paid equally. | 60 | |
omni_math | [
{
"content": "Kimothy starts in the bottom-left square of a 4 by 4 chessboard. In one step, he can move up, down, left, or right to an adjacent square. Kimothy takes 16 steps and ends up where he started, visiting each square exactly once (except for his starting/ending square). How many paths could he have tak... | {
"ground_truth": "12",
"style": "rule"
} | Kimothy starts in the bottom-left square of a 4 by 4 chessboard. In one step, he can move up, down, left, or right to an adjacent square. Kimothy takes 16 steps and ends up where he started, visiting each square exactly once (except for his starting/ending square). How many paths could he have taken? | 12 | 12 |
deepscaler | [
{
"content": "Determine the remainder when $$\\sum_{i=0}^{2015}\\left\\lfloor\\frac{2^{i}}{25}\\right\\rfloor$$ is divided by 100, where $\\lfloor x\\rfloor$ denotes the largest integer not greater than $x$.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | Determine the remainder when $$\sum_{i=0}^{2015}\left\lfloor\frac{2^{i}}{25}\right\rfloor$$ is divided by 100, where $\lfloor x\rfloor$ denotes the largest integer not greater than $x$. | 14 | Let $r_{i}$ denote the remainder when $2^{i}$ is divided by 25. Note that because $2^{\phi(25)} \equiv 2^{20} \equiv 1(\bmod 25)$, $r$ is periodic with length 20. In addition, we find that 20 is the order of $2 \bmod 25$. Since $2^{i}$ is never a multiple of 5, all possible integers from 1 to 24 are represented by $r_{... |
deepscaler | [
{
"content": "Let the function $f(x)=2x-\\cos x$, and $\\{a_n\\}$ be an arithmetic sequence with a common difference of $\\dfrac{\\pi}{8}$. If $f(a_1)+f(a_2)+\\ldots+f(a_5)=5\\pi$, calculate $\\left[f(a_3)\\right]^2-a_2a_3$.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{13}{16}\\pi^2",
"style": "rule"
} | Let the function $f(x)=2x-\cos x$, and $\{a_n\}$ be an arithmetic sequence with a common difference of $\dfrac{\pi}{8}$. If $f(a_1)+f(a_2)+\ldots+f(a_5)=5\pi$, calculate $\left[f(a_3)\right]^2-a_2a_3$. | \dfrac{13}{16}\pi^2 | |
deepscaler | [
{
"content": "Find the product of all constants \\(t\\) such that the quadratic \\(x^2 + tx + 12\\) can be factored in the form \\((x+a)(x+b)\\), where \\(a\\) and \\(b\\) are integers.",
"role": "user"
}
] | {
"ground_truth": "-530,784",
"style": "rule"
} | Find the product of all constants \(t\) such that the quadratic \(x^2 + tx + 12\) can be factored in the form \((x+a)(x+b)\), where \(a\) and \(b\) are integers. | -530,784 | |
math_dapo | [
{
"content": "Two congruent right circular cones each with base radius $3$ and height $8$ have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $3$ from the base of each cone. A sphere with radius $r$ lies within both cones. The maximum possible value of $r^... | {
"ground_truth": "298",
"style": "rule-lighteval/MATH_v2"
} | Two congruent right circular cones each with base radius $3$ and height $8$ have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $3$ from the base of each cone. A sphere with radius $r$ lies within both cones. The maximum possible value of $r^2$ is $\frac{m}{n}$, w... | 298 | 298 |
amc | [
{
"content": "The diagram shows an octagon consisting of $10$ unit squares. The portion below $\\overline{PQ}$ is a unit square and a triangle with base $5$. If $\\overline{PQ}$ bisects the area of the octagon, what is the ratio $\\dfrac{XQ}{QY}$?",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | The diagram shows an octagon consisting of $10$ unit squares. The portion below $\overline{PQ}$ is a unit square and a triangle with base $5$. If $\overline{PQ}$ bisects the area of the octagon, what is the ratio $\dfrac{XQ}{QY}$? | \frac{2}{3} | \frac{2}{3} |
deepscaler | [
{
"content": "The sum of the digits of the positive integer $N$ is three times the sum of the digits of $N+1$. What is the smallest possible sum of the digits of $N$?",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | The sum of the digits of the positive integer $N$ is three times the sum of the digits of $N+1$. What is the smallest possible sum of the digits of $N$? | 12 | |
math_dapo | [
{
"content": "What is the hundreds digit of $(20! - 15!)$?",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | What is the hundreds digit of $(20! - 15!)$? | 0 | 0 |
deepscaler | [
{
"content": "If $x$, $y$, and $z$ are positive numbers satisfying \\[\nx+\\frac{1}{y}=4,\\ \\ \\ y+\\frac{1}{z}=1,\\text{ and }z+\\frac{1}{x}=\\frac{7}{3},\n\\]find the value of $xyz$.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | If $x$, $y$, and $z$ are positive numbers satisfying \[
x+\frac{1}{y}=4,\ \ \ y+\frac{1}{z}=1,\text{ and }z+\frac{1}{x}=\frac{7}{3},
\]find the value of $xyz$. | 1 | |
deepscaler | [
{
"content": "\nA two-digit integer between 10 and 99, inclusive, is chosen at random. Each possible integer is equally likely to be chosen. What is the probability that its tens digit is a multiple of its units (ones) digit?",
"role": "user"
}
] | {
"ground_truth": "23/90",
"style": "rule"
} |
A two-digit integer between 10 and 99, inclusive, is chosen at random. Each possible integer is equally likely to be chosen. What is the probability that its tens digit is a multiple of its units (ones) digit? | 23/90 | |
deepscaler | [
{
"content": "Find the maximum value of the parameter \\( b \\) for which the inequality \\( b \\sqrt{b}\\left(x^{2}-10 x+25\\right)+\\frac{\\sqrt{b}}{\\left(x^{2}-10 x+25\\right)} \\leq \\frac{1}{5} \\cdot \\sqrt[4]{b^{3}} \\cdot \\left| \\sin \\frac{\\pi x}{10} \\right| \\) has at least one solution.",
"r... | {
"ground_truth": "1/10000",
"style": "rule"
} | Find the maximum value of the parameter \( b \) for which the inequality \( b \sqrt{b}\left(x^{2}-10 x+25\right)+\frac{\sqrt{b}}{\left(x^{2}-10 x+25\right)} \leq \frac{1}{5} \cdot \sqrt[4]{b^{3}} \cdot \left| \sin \frac{\pi x}{10} \right| \) has at least one solution. | 1/10000 | |
deepscaler | [
{
"content": "Grisha has 5000 rubles. Chocolate bunnies are sold in a store at a price of 45 rubles each. To carry the bunnies home, Grisha will have to buy several bags at 30 rubles each. One bag can hold no more than 30 chocolate bunnies. Grisha bought the maximum possible number of bunnies and enough bags to... | {
"ground_truth": "20",
"style": "rule"
} | Grisha has 5000 rubles. Chocolate bunnies are sold in a store at a price of 45 rubles each. To carry the bunnies home, Grisha will have to buy several bags at 30 rubles each. One bag can hold no more than 30 chocolate bunnies. Grisha bought the maximum possible number of bunnies and enough bags to carry all the bunnies... | 20 | |
aime | [
{
"content": "There are $n$ mathematicians seated around a circular table with $n$ seats numbered $1,$ $2,$ $3,$ $...,$ $n$ in clockwise order. After a break they again sit around the table. The mathematicians note that there is a positive integer $a$ such that\n ($1$) for each $k,$ the mathematician who was se... | {
"ground_truth": "332",
"style": "rule"
} | There are $n$ mathematicians seated around a circular table with $n$ seats numbered $1,$ $2,$ $3,$ $...,$ $n$ in clockwise order. After a break they again sit around the table. The mathematicians note that there is a positive integer $a$ such that
($1$) for each $k,$ the mathematician who was seated in seat $k$ before... | 332 | 332 |
math | [
{
"content": "For $\\{1, 2, 3, \\ldots, n\\}$ and each of its non-empty subsets a unique alternating sum is defined as follows. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. For example, the alternating sum for $\\{1,... | {
"ground_truth": "448",
"style": "rule"
} | For $\{1, 2, 3, \ldots, n\}$ and each of its non-empty subsets a unique alternating sum is defined as follows. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. For example, the alternating sum for $\{1, 2, 3, 6,9\}$ is $9-6+3-2+... | 448 | 448 |
deepscaler | [
{
"content": "What is the maximum number of queens that can be placed on an $8 \\times 8$ chessboard so that each queen can attack at least one other queen?",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | What is the maximum number of queens that can be placed on an $8 \times 8$ chessboard so that each queen can attack at least one other queen? | 16 | |
deepscaler | [
{
"content": "Let $a_n = -n^2 + 10n + 11$, then find the value of $n$ for which the sum of the sequence $\\{a_n\\}$ from the first term to the nth term is maximized.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | Let $a_n = -n^2 + 10n + 11$, then find the value of $n$ for which the sum of the sequence $\{a_n\}$ from the first term to the nth term is maximized. | 11 | |
deepscaler | [
{
"content": "Add together the numbers $7.56$ and $4.29$. Write the result as a decimal.",
"role": "user"
}
] | {
"ground_truth": "11.85",
"style": "rule"
} | Add together the numbers $7.56$ and $4.29$. Write the result as a decimal. | 11.85 | |
deepscaler | [
{
"content": "Paul fills in a $7 \\times 7$ grid with the numbers 1 through 49 in a random arrangement. He then erases his work and does the same thing again (to obtain two different random arrangements of the numbers in the grid). What is the expected number of pairs of numbers that occur in either the same ro... | {
"ground_truth": "147 / 2",
"style": "rule"
} | Paul fills in a $7 \times 7$ grid with the numbers 1 through 49 in a random arrangement. He then erases his work and does the same thing again (to obtain two different random arrangements of the numbers in the grid). What is the expected number of pairs of numbers that occur in either the same row as each other or the ... | 147 / 2 | Each of the $\binom{49}{2}$ pairs of numbers has a probability of $\frac{14 \cdot\binom{7}{2}}{\binom{49}{2}}=1 / 4$ of being in the same row or column in one of the arrangements, so the expected number that are in the same row or column in both arrangements is $$\binom{49}{2} \cdot(1 / 4)^{2}=\frac{147}{2}$$ |
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