data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Compute $(4+8)^2+(4^2+8^2)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "224"
} | {
"split": "train",
"index": 36849,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the arithmetic sequence $\\{a_{n}\\}$, if $\\frac{{a}_{9}}{{a}_{8}}＜-1$, and its sum of the first $n$ terms $S_{n}$ has a minimum value, determine the minimum value of $n$ for which $S_{n} \\gt 0$. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 36850,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Say that an integer $B$ is yummy if there exist several consecutive integers, including $B$, that add up to 2023. What is the smallest yummy integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2022"
} | {
"split": "train",
"index": 36851,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The points $(0,0)\\,$, $(a,11)\\,$, and $(b,37)\\,$ are the vertices of an equilateral triangle. Find the value of $ab\\,$.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "315"
} | {
"split": "train",
"index": 36852,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Martha can make 24 cookies with 3 cups of flour. How many cookies can she make with 5 cups of flour, and how many cups of flour are needed to make 60 cookies? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7.5"
} | {
"split": "train",
"index": 36853,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiao Ming throws a die with uniform density three times and observes the number of points on the upper face each time. It is known that the numbers of points in the three throws are all different. Calculate the probability that the sum of the three numbers of points does not ex... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"split": "train",
"index": 36854,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Complex numbers $p,$ $q,$ $r$ form an equilateral triangle with side length 24 in the complex plane. If $|p + q + r| = 48,$ find $|pq + pr + qr|.$ These complex numbers have been translated by the same complex number $z$ compared to their original positions on the origin. Let's... | math | {
"style": "rule",
"ground_truth": "768"
} | {
"split": "train",
"index": 36855,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$, $ADE$, and $EFG$ are all equilateral. Points $D$ and $G$ are midpoints of $\\overline{AC}$ and $\\overline{AE}$, respectively. If $AB=4$, what is the perimeter of figure $ABCDEFG$? [asy]\n/* AMC8 2000 #15 Problem */\ndraw((0,0)--(4,0)--(5,2)--(5.5,1)--(4.5,1));\... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 36856,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The letter T is formed by placing a $2\\:\\text{inch} \\times 6\\:\\text{inch}$ rectangle vertically and a $2\\:\\text{inch} \\times 4\\:\\text{inch}$ rectangle horizontally across the top center of the vertical rectangle. What is the perimeter of the T, in inches? Let's think ... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 36857,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $[(12^{12} \\div 12^{11})^2 \\cdot 4^2] \\div 2^4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 36858,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x$ and $y$ are positive integers less than $30$ for which $x + y + xy = 104$, what is the value of $x + y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 36859,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, the equation of curve $C_{1}$ is $x^{2}+y^{2}=1$. Taking the origin $O$ of the Cartesian coordinate system $xOy$ as the pole and the positive half of the $x$-axis as the polar axis, a polar coordinate system is established with the same... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt {5}"
} | {
"split": "train",
"index": 36860,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $a$ and $b$ are positive numbers such that $a^b = b^a$ and $b = 4a$, solve for the value of $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[3]{4}"
} | {
"split": "train",
"index": 36861,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "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 $\\angle ACB.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75^\\circ"
} | {
"split": "train",
"index": 36862,
"solution": "1. **Given Information and Setup:**\n - In $\\triangle ABC$, $\\angle ABC = 45^\\circ$.\n - Point $D$ is on $\\overline{BC}$ such that $2 \\cdot BD = CD$.\n - $\\angle DAB = 15^\\circ$.\n - Let $\\angle ACB = \\theta$.\n\n2. **Using the Angle Bisector Theor... |
deepscaler | [
{
"role": "user",
"content": "A can is in the shape of a right circular cylinder. The circumference of the base of the can is 12 inches, and the height of the can is 5 inches. A spiral strip is painted on the can in such a way that it winds around the can exactly once as it reaches from the bottom of the ca... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 36863,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "You know that the Jones family has five children, and the Smith family has three children. Of the eight children you know that there are five girls and three boys. Let $\\dfrac{m}{n}$ be the probability that at least one of the families has only girls for children. Given ... | math | {
"style": "rule",
"ground_truth": "67"
} | {
"split": "train",
"index": 36864,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The greatest common divisor of two integers is $(x+3)$ and their least common multiple is $x(x+3)$, where $x$ is a positive integer. If one of the integers is 36, what is the smallest possible value of the other one? Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 36865,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using the digits 0, 1, 2, 3, 4, 5 to form numbers without repeating any digit. Calculate: \n(1) How many six-digit numbers can be formed? \n(2) How many three-digit numbers can be formed that contain at least one even number? \n(3) How many three-digit numbers can be formed ... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 36866,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A courier set out on a moped from city $A$ to city $B$, which are 120 km apart. One hour later, a second courier on a motorcycle left from $A$. The second courier caught up to the first, delivered a message, and immediately returned to $A$ at the same speed, arriving back in $A... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 36867,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Elizabetta wants to write the integers 1 to 9 in the regions of the shape shown, with one integer in each region. She wants the product of the integers in any two regions that have a common edge to be not more than 15. In how many ways can she do this? Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 36868,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, \\(\\triangle ABC\\) and \\(\\triangle CDE\\) are equilateral triangles. Given that \\(\\angle EBD = 62^\\circ\\) and \\(\\angle AEB = x^\\circ\\), what is the value of \\(x\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "122"
} | {
"split": "train",
"index": 36869,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, the rectangle has a width $w$, a length of $8$, and a perimeter of $24$. What is the ratio of its width to its length? [asy]\npair a = (0, 0); pair b = (8, 0); pair c = (8, 4); pair d = (0, 4);\ndraw(a--b--c--d--cycle);\nlabel(\"$w$\", midpoint(a--d), W); label(... | math | {
"style": "rule",
"ground_truth": "1 : 2"
} | {
"split": "train",
"index": 36870,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a polynomial with integer coefficients given by:\n\\[8x^5 + b_4 x^4 + b_3 x^3 + b_2 x^2 + b_1 x + 24 = 0.\\]\nFind the number of different possible rational roots of this polynomial. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 36871,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $(x+a)(x+8)=x^{2}+bx+24$ for all values of $x$, what is the value of $a+b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 36872,
"solution": "Since $(x+a)(x+8)=x^{2}+bx+24$ for all $x$, then $x^{2}+ax+8x+8a=x^{2}+bx+24$ or $x^{2}+(a+8)x+8a=x^{2}+bx+24$ for all $x$. Since the equation is true for all $x$, then the coefficients on the left side must match the coefficients on the right side. Therefore, $a+8... |
deepscaler | [
{
"role": "user",
"content": "If $\\log_{10}2=a$ and $\\log_{10}3=b$, then $\\log_{5}12=?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2a+b}{1-a}"
} | {
"split": "train",
"index": 36873,
"solution": "1. **Identify the given values and the target expression**: We are given $\\log_{10}2=a$ and $\\log_{10}3=b$. We need to find $\\log_{5}12$.\n\n2. **Use the change of base formula**: According to the change of base formula, for any logarithms $\\log_b a = \\frac{\\... |
deepscaler | [
{
"role": "user",
"content": "The integers \\(1,2,3,4,5,6,7,8,9,10\\) are written on a blackboard. Each day, a teacher chooses one of the integers uniformly at random and decreases it by 1. Let \\(X\\) be the expected value of the number of days which elapse before there are no longer positive integers on t... | math | {
"style": "rule",
"ground_truth": "120.75280458176904"
} | {
"split": "train",
"index": 36874,
"solution": "Answer: 120.75280458176904"
} |
deepscaler | [
{
"role": "user",
"content": "Given that the product of the digits of a 3-digit positive integer equals 36, calculate the number of such integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 36875,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\[p(x,y) =\n\\begin{cases} x + y &\\quad \\text{if } x \\ge 0 \\text{ and } y \\ge 0, \\\\\nx - 2y &\\quad \\text{if } x < 0 \\text{ and } y < 0, \\\\\n3x + y &\\quad \\text{otherwise}.\n\\end{cases}\n\\]What is $p(p(1,-1),p(-5,-2))$? Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 36876,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest number written using only the digits 0 and 1 that is divisible by the product of the six smallest natural numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1111111110000"
} | {
"split": "train",
"index": 36877,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a function $f(x)$ that satisfies: For any $x \\in (0, +\\infty)$, it always holds that $f(2x) = 2f(x)$; (2) When $x \\in (1, 2]$, $f(x) = 2 - x$. If $f(a) = f(2020)$, find the smallest positive real number $a$. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 36878,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many rational terms are in the expansion of\n\na) $(\\sqrt{2}+\\sqrt[4]{3})^{100}$\n\nb) $(\\sqrt{2}+\\sqrt[3]{3})^{300}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 36879,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different positions can appear on a chessboard if both players, starting from the initial position, make just one move each? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 36880,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rectangle $ABCD$ has $AB=4$ and $BC=3$. Segment $EF$ is constructed through $B$ so that $EF$ is perpendicular to $DB$, and $A$ and $C$ lie on $DE$ and $DF$, respectively. What is $EF$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{125}{12}"
} | {
"split": "train",
"index": 36881,
"solution": "\n#### Step 1: Understanding the Problem\nWe are given a rectangle $ABCD$ with $AB = 4$ and $BC = 3$. A segment $EF$ is constructed through $B$ such that $EF$ is perpendicular to $DB$, and points $A$ and $C$ lie on $DE$ and $DF$, respectively. We need to find the l... |
deepscaler | [
{
"role": "user",
"content": "Given that 28×15=420, directly write out the results of the following multiplications:\n2.8×1.5=\\_\\_\\_\\_\\_\\_、0.28×1.5=\\_\\_\\_\\_\\_\\_、0.028×0.15=\\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.0042"
} | {
"split": "train",
"index": 36882,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sides of a right-angled triangle are positive integers, and the perimeter of the triangle is equal to the area of the triangle, find the length of the hypotenuse. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 36883,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Polygon $ABCDEF$ is a regular hexagon. What is the measure in degrees of angle $ABF$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 36884,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In parallelogram $ABCD$, line $DP$ is drawn bisecting $BC$ at $N$ and meeting $AB$ (extended) at $P$. From vertex $C$, line $CQ$ is drawn bisecting side $AD$ at $M$ and meeting $AB$ (extended) at $Q$. Lines $DP$ and $CQ$ meet at $O$. If the area of parallelogram $ABCD$ is $k$, ... | math | {
"style": "rule",
"ground_truth": "\\frac{9k}{8}"
} | {
"split": "train",
"index": 36885,
"solution": "1. **Identify Key Points and Relationships**: In parallelogram $ABCD$, $DP$ bisects $BC$ at $N$ and extends to meet $AB$ at $P$. Similarly, $CQ$ bisects $AD$ at $M$ and extends to meet $AB$ at $Q$. Lines $DP$ and $CQ$ intersect at point $O$.\n\n2. **Area Decomposit... |
deepscaler | [
{
"role": "user",
"content": "A shape is created by aligning five unit cubes in a straight line. Then, one additional unit cube is attached to the top of the second cube in the line and another is attached beneath the fourth cube in the line. Calculate the ratio of the volume to the surface area. Let's thin... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 36886,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\frac{x}{w}$ if $w \\neq 0$ and $\\frac{x+6 y-3 z}{-3 x+4 w}=\\frac{-2 y+z}{x-w}=\\frac{2}{3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 36887,
"solution": "We have $x+6 y-3 z=\\frac{2}{3}(-3 x+4 w)$ and $-2 y+z=\\frac{2}{3}(x-w)$, so $$\\frac{x}{w}=\\frac{(x+6 y-3 z)+3(-2 y+z)}{(-3 x+4 w)+3(x-w)}=\\frac{\\frac{2}{3}(-3 x+4 w)+3 \\cdot \\frac{2}{3}(x-w)}{(-3 x+4 w)+3(x-w)}=\\frac{\\frac{2}{3}[(-3 x+4 w)+3(x-w)]}{(-3 x+... |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\vec{a}$ and $\\vec{b}$ with magnitudes $|\\vec{a}|=2$ and $|\\vec{b}|=\\sqrt{3}$, respectively, and the equation $( \\vec{a}+2\\vec{b}) \\cdot ( \\vec{b}-3\\vec{a})=9$:\n\n(1) Find the dot product $\\vec{a} \\cdot \\vec{b}$.\n\n(2) In triangle $ABC$, with $\\ve... | math | {
"style": "rule",
"ground_truth": "-\\sqrt{3}"
} | {
"split": "train",
"index": 36888,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point A is a fixed point on a circle with a circumference of 3. If a point B is randomly selected on the circumference of the circle, the probability that the length of the minor arc is less than 1 is ______. Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 36889,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sum_{k=1}^{40}\\sin 4k=\\tan \\frac{p}{q},$ where angles are measured in degrees, and $p$ and $q$ are relatively prime positive integers that satisfy $\\frac{p}{q}<90,$ find $p+q.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "85"
} | {
"split": "train",
"index": 36890,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\sin 270^\\circ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 36891,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\begin{vmatrix} 7 & 3 \\\\ -1 & 2 \\end{vmatrix}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 36892,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $Z$ denote the set of points in $\\mathbb{R}^n$ whose coordinates are 0 or 1. (Thus $Z$ has $2^n$ elements, which are the vertices of a unit hypercube in $\\mathbb{R}^n$.) Given a vector subspace $V$ of $\\mathbb{R}^n$, let $Z(V)$ denote the number of members of $Z$ that li... | math | {
"style": "rule",
"ground_truth": "2^k"
} | {
"split": "train",
"index": 36893,
"solution": "The maximum is $2^k$, achieved for instance by the subspace \\[\\{(x_1, \\dots, x_n) \\in \\mathbb{R}^n: x_1 = \\cdots = x_{n-k} = 0\\}.\\]\n\n\\textbf{First solution:} More generally, we show that any affine $k$-dimensional plane in $\\mathbb{R}^n$ can contain at ... |
deepscaler | [
{
"role": "user",
"content": "Let set \\( A = \\{a \\mid a = 3k + 2, k \\leqslant 2000, k \\in \\mathbf{N}_{+}\\} \\) and set \\( B = \\{b \\mid b = 4k - 1, k \\leqslant 2000, k \\in \\mathbf{N}_{+}\\} \\). How many elements are in \\( A \\cap B \\)? Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "500"
} | {
"split": "train",
"index": 36894,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ming and Catherine walked to the store together to buy some pencils. Ming bought $40$ pencils and Catherine bought $24$. If each package of pencils sold at the store contain the same number of pencils, what is the largest possible number of pencils in a package? Let's think s... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 36895,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For the specific example $M=5$, find a value of $k$, not necessarily the smallest, such that $\\sum_{n=1}^{k} \\frac{1}{n}>M$. Justify your answer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "256"
} | {
"split": "train",
"index": 36896,
"solution": "Note that $\\frac{1}{n+1}+\\frac{1}{n+2}+\\ldots+\\frac{1}{2n}>\\frac{1}{2n}+\\ldots+\\frac{1}{2n}=\\frac{1}{2}$. Therefore, if we apply this to $n=1,2,4,8,16,32,64,128$, we get $\\left(\\frac{1}{2}\\right)+\\left(\\frac{1}{3}+\\frac{1}{4}\\right)+\\left(\\frac{1}{... |
deepscaler | [
{
"role": "user",
"content": "Let the function $y=f(k)$ be a monotonically increasing function defined on $N^*$, and $f(f(k))=3k$. Find the value of $f(1)+f(9)+f(10)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 36897,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( |z|=2 \\) and \\( u=\\left|z^{2}-z+1\\right| \\), find the minimum value of \\( u \\) where \\( z \\in \\mathbf{C} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2} \\sqrt{3}"
} | {
"split": "train",
"index": 36898,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $a * b$ means $3a-b.$ What is the value of $x$ if $2 * (5 * x)=1$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 36899,
"solution": "1. **Interpret the operation $*$**: Given that $a * b = 3a - b$, we need to find $x$ such that $2 * (5 * x) = 1$.\n\n2. **Evaluate $5 * x$**: According to the operation definition, $5 * x = 3 \\cdot 5 - x = 15 - x$.\n\n3. **Substitute $5 * x$ into the equation**: N... |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be a set of size $11$ . A random $12$ -tuple $(s_1, s_2, . . . , s_{12})$ of elements of $S$ is chosen uniformly at random. Moreover, let $\\pi : S \\to S$ be a permutation of $S$ chosen uniformly at random. The probability that $s_{i+1}\\ne \\pi (s_i)$ fo... | math | {
"style": "rule",
"ground_truth": "1000000000004"
} | {
"split": "train",
"index": 36900,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of games won by six volleyball teams are displayed in a graph, but the names of the teams are missing. The following clues provide information about the teams:\n\n1. The Falcons won more games than the Hawks.\n2. The Warriors won more games than the Knights but fewer... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 36901,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $w$ is a complex number such that $w^2 = 45-21i$. Find $|w|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[4]{2466}"
} | {
"split": "train",
"index": 36902,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of triangle $ABC$ below.\n\n[asy]\nunitsize(1inch);\npair A, B, C;\nA = (0,0);\nB= (sqrt(2),0);\nC = (0,sqrt(2));\ndraw (A--B--C--A, linewidth(0.9));\ndraw(rightanglemark(B,A,C,3));\nlabel(\"$A$\",A,S);\nlabel(\"$B$\",B,S);\nlabel(\"$C$\",C,N);\nlabel(\"$8$\",(B+C... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 36903,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $$\\lceil\\sqrt{5}\\rceil + \\lceil\\sqrt{6}\\rceil + \\lceil\\sqrt{7}\\rceil + \\cdots + \\lceil\\sqrt{49}\\rceil$$Note: For a real number $x,$ $\\lceil x \\rceil$ denotes the smallest integer that is greater than or equal to $x.$ Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "245"
} | {
"split": "train",
"index": 36904,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A semipro baseball league has teams with $21$ players each. League rules state that a player must be paid at least $15,000$ dollars, and that the total of all players' salaries for each team cannot exceed $700,000$ dollars. What is the maximum possible salary, in dollars, for a... | math | {
"style": "rule",
"ground_truth": "400000"
} | {
"split": "train",
"index": 36905,
"solution": "1. **Understand the constraints**: Each team has $21$ players. Each player must be paid at least $15,000$ dollars. The total salary for all players on a team cannot exceed $700,000$ dollars.\n\n2. **Set up the equation**: To find the maximum possible salary for a s... |
deepscaler | [
{
"role": "user",
"content": "Convert the binary number $11011001_2$ to base 4. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3121_4"
} | {
"split": "train",
"index": 36906,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle JKL$ shown, $\\tan K = \\frac{3}{2}$. What is $KL$?\n\n[asy]\npair J,K,L;\nL = (0,0);\nJ = (0,3);\nK = (2,3);\ndraw(L--J--K--L);\ndraw(rightanglemark(L,J,K,7));\nlabel(\"$L$\",L,SW);\nlabel(\"$J$\",J,NW);\nlabel(\"$K$\",K,NE);\nlabel(\"$2$\",(J+K)/2,N);\... | math | {
"style": "rule",
"ground_truth": "\\sqrt{13}"
} | {
"split": "train",
"index": 36907,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Pompous Vova has an iPhone XXX, and on that iPhone, he has a calculator with voice commands: \"Multiply my number by two and subtract two from the result,\" \"Multiply my number by three and then add four,\" and lastly, \"Add seven to my number!\" The iPhone knows that initiall... | math | {
"style": "rule",
"ground_truth": "9000"
} | {
"split": "train",
"index": 36908,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a triangle with vertices at points \\( (0, 0), (30, 0), \\) and \\( (18, 26) \\). The vertices of its midpoint triangle are the midpoints of its sides. A triangular pyramid is formed by folding the original triangle along the sides of its midpoint triangle. What is the... | math | {
"style": "rule",
"ground_truth": "3380"
} | {
"split": "train",
"index": 36909,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the game of preference, each of the three players is dealt 10 cards, and two cards are placed in the kitty. How many different arrangements are possible in this game? (Consider possible distributions without accounting for which 10 cards go to each specific player.) Let's th... | math | {
"style": "rule",
"ground_truth": "\\frac{32!}{(10!)^3 \\cdot 2! \\cdot 3!}"
} | {
"split": "train",
"index": 36910,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest eight-digit positive integer that has exactly four digits which are 4? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10004444"
} | {
"split": "train",
"index": 36911,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The value of\n\\[\\frac{n}{2} + \\frac{18}{n}\\]is smallest for which positive integer $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 36912,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\{a_n\\}$ is an arithmetic sequence with common difference $d$, and $S_n$ is the sum of the first $n$ terms, if only $S_4$ is the minimum term among $\\{S_n\\}$, then the correct conclusion(s) can be drawn is/are ______.\n\n$(1) d > 0$ $(2) a_4 < 0$ $(3) a_5 >... | math | {
"style": "rule",
"ground_truth": "(1)(2)(3)(4)"
} | {
"split": "train",
"index": 36913,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A stack of $2000$ cards is labelled with the integers from $1$ to $2000,$ with different integers on different cards. The cards in the stack are not in numerical order. The top card is removed from the stack and placed on the table, and the next card is moved to the bottom of t... | math | {
"style": "rule",
"ground_truth": "927"
} | {
"split": "train",
"index": 36914,
"solution": "To simplify matters, we want a power of $2$. Hence, we will add $48$ 'fake' cards which we must discard in our actual count. Using similar logic as Solution 1, we find that 1999 has position $1024$ in a $2048$ card stack, where the fake cards towards the front.\n L... |
deepscaler | [
{
"role": "user",
"content": "Bernardo and Silvia play the following game. An integer between $0$ and $999$ inclusive is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds $50$ to it and passes the r... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 36915,
"solution": "\nWe need to find the smallest initial number $N$ such that Bernardo wins the game. The game involves Bernardo doubling the number and Silvia adding 50 to it. The sequence of operations will continue until the number exceeds 999.\n\n#### Step-by-step Analysis:\n\n1... |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, the parametric equations of curve $C_{1}$ are $\\left\\{\\begin{array}{l}{x=2+2\\cos\\varphi}\\\\{y=2\\sin\\varphi}\\end{array}\\right.$ ($\\varphi$ is the parameter). Taking the origin $O$ as the pole and the positive half of the $x$-a... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\pi}{4}"
} | {
"split": "train",
"index": 36916,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To address the threat of AIDS to humanity, now three research institutes, designated as Institute A, Institute B, and Institute C, are independently developing an AIDS vaccine. The probabilities of successfully developing a vaccine are respectively $\\frac{1}{2}$, $\\frac{1}{3}... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 36917,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let sets \\( A \\) and \\( B \\) satisfy:\n\\[ \nA \\cup B = \\{1, 2, \\cdots, 10\\}, \\quad A \\cap B = \\varnothing \n\\]\nIf the number of elements in set \\( A \\) is not an element of \\( A \\) and the number of elements in set \\( B \\) is not an element of \\( B \\), fin... | math | {
"style": "rule",
"ground_truth": "186"
} | {
"split": "train",
"index": 36918,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( X \\), \\( Y \\), and \\( Z \\) be nonnegative integers such that \\( X+Y+Z = 15 \\). What is the maximum value of \n\\[ X \\cdot Y \\cdot Z + X \\cdot Y + Y \\cdot Z + Z \\cdot X? \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 36919,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle passes through the vertices of a triangle with side-lengths $7\\tfrac{1}{2},10,12\\tfrac{1}{2}.$ The radius of the circle is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{25}{4}"
} | {
"split": "train",
"index": 36920,
"solution": "1. **Identify the Type of Triangle**: \n Given the side lengths of the triangle are $7\\tfrac{1}{2}$, $10$, and $12\\tfrac{1}{2}$. We first convert these mixed fractions to improper fractions for easier calculation:\n \\[\n 7\\tfrac{1}{2} = \\frac{15}{2}, \\... |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, with $BC=15$, $AC=10$, and $\\angle A=60^\\circ$, find $\\cos B$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{3}"
} | {
"split": "train",
"index": 36921,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $b_n$ be the integer obtained by writing down the integers from 1 to $n$ in reverse, from right to left. Compute the remainder when $b_{39}$ is divided by 125. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 36922,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integer pairs $(x,y)$ are there such that \\[0\\leq x < 165, \\quad 0\\leq y < 165 \\text{ and } y^2\\equiv x^3+x \\pmod {165}?\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "99"
} | {
"split": "train",
"index": 36923,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( x^{2} + y^{2} - 2x - 2y + 1 = 0 \\) where \\( x, y \\in \\mathbb{R} \\), find the minimum value of \\( F(x, y) = \\frac{x + 1}{y} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3/4"
} | {
"split": "train",
"index": 36924,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Lara is trying to solve the following equation by completing the square: $$100x^2 + 60x - 49 = 0.$$ She seeks to rewrite the above equation in the form: $$(ax + b)^2 = c,$$ where $a$, $b$, and $c$ are integers and $a > 0$. What is the value of $a + b + c$? Let's think step by s... | math | {
"style": "rule",
"ground_truth": "71"
} | {
"split": "train",
"index": 36925,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In Montana, 500 people were asked what they call soft drinks. The results of the survey are shown in the pie chart. The central angle of the ``Soda'' sector of the graph is $200^\\circ$, to the nearest whole degree. How many of the people surveyed chose ``Soda''? Express your a... | math | {
"style": "rule",
"ground_truth": "278"
} | {
"split": "train",
"index": 36926,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jordan and his three best friends are on a relay team. His relay team will run a race, where the first runner runs a lap, then the second, then the third, then the fourth.\n\nJordan is the fastest so he will run the fourth lap. How many different orders can the four team memb... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 36927,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The following is the process of simplifying fractions by Xiaobai. Please read carefully and complete the corresponding tasks.\n\nSolution: $(\\frac{3x+4}{x^2-1}-\\frac{2}{x-1})÷\\frac{x+2}{x^2-2x+1}$\n\n$=[\\frac{3x+4}{(x+1)(x-1)}-\\frac{2}{x-1}]÷\\frac{x+2}{(x-1)^2}\\ldots \\l... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 36928,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $A, B$ and $C$ lie on the same line so that $CA = AB$ . Square $ABDE$ and the equilateral triangle $CFA$ , are constructed on the same side of line $CB$ . Find the acute angle between straight lines $CE$ and $BF$ . Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 36929,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse and a hyperbola,\\[\\frac{x^2}{16} - \\frac{y^2}{25} = 1\\]and \\[\\frac{x^2}{K} + \\frac{y^2}{25} = 1\\], have the same asymptotes. Find the value of $K$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 36930,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(In this question, 12 points) During a shooting training session, the probabilities of a shooter hitting the 10, 9, 8, and 7 rings are 0.21, 0.23, 0.25, and 0.28, respectively. Calculate the probability that the shooter in a single shot:\n(1) Hits either the 10 or 7 ring; \n(2... | math | {
"style": "rule",
"ground_truth": "0.03"
} | {
"split": "train",
"index": 36931,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A horse 24 feet from the center of a merry-go-round makes 32 revolutions. In order to travel the same distance, how many revolutions would a horse 8 feet from the center have to make? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 36932,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The increasing sequence \\(1, 3, 4, 9, 10, 12, 13, \\cdots\\) consists of some positive integers that are either powers of 3 or sums of distinct powers of 3. Find the value of the 2014th term. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "88329"
} | {
"split": "train",
"index": 36933,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the point \\( P \\) on the ellipse \\(\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1 \\left(a>b>0, c=\\sqrt{a^{2}-b^{2}}\\right)\\), and the equation of the line \\( l \\) is \\(x=-\\frac{a^{2}}{c}\\), and the coordinate of the point \\( F \\) is \\((-c, 0)\\). Draw \\( PQ \... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 36934,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the largest prime factor of the sum \\(\\sum_{k=1}^{11} k^{5}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "263"
} | {
"split": "train",
"index": 36935,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all integer values of \\( c \\) for which the equation \\( 15|p-1| + |3p - |p + c|| = 4 \\) has at least one solution for \\( p \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 36936,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:00 AM, and all three always take the same amount of time to eat lunch. On Monday the three of them painted 50% of a house, quitting at 4:00 PM. On Tuesday, when Paula wasn... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 36937,
"solution": "1. **Define Variables:**\n Let $p$ be the rate at which Paula paints (in house/hours), $h$ be the combined rate of the two helpers (in house/hours), and $L$ be the duration of the lunch break (in hours).\n\n2. **Set Up Equations:**\n From the problem, we can es... |
deepscaler | [
{
"role": "user",
"content": "The maximum value and the minimum positive period of the function $f(x)=\\cos 4x \\cdot \\cos 2x \\cdot \\cos x \\cdot \\sin x$ are to be determined. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 36938,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "150 people were surveyed and asked: \"Do you think teal is more green or blue?\" Of them, 90 believe teal is \"more green,\" and 50 believe it's \"more blue.\" Additionally, 40 believe it's both \"more green\" and \"more blue.\" Another 20 think teal is neither \"more green\" n... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 36939,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers \\(n\\) satisfy \\((n+5)(n-9) \\le 0\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 36940,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\sum_{n = 2}^{10000} \\frac{1}{n \\sqrt{n - 1} + (n - 1) \\sqrt{n}}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{99}{100}"
} | {
"split": "train",
"index": 36941,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\frac{x^8+12x^4+36}{x^4+6}$ when $x=5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "631"
} | {
"split": "train",
"index": 36942,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Vasya wrote a set of distinct natural numbers on the board, each of which does not exceed 2023. It turned out that for any two written numbers \\(a\\) and \\(b\\), the number \\(a + b\\) is not divisible by the number \\(a - b\\). What is the maximum number of numbers Vasya mig... | math | {
"style": "rule",
"ground_truth": "675"
} | {
"split": "train",
"index": 36943,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( \\triangle ABC \\), point \\( D \\) is on side \\( BC \\) such that \\( AD \\perp BC \\) and \\( AD = BC = a \\). Find the maximum value of \\( \\frac{b}{c} + \\frac{c}{b} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2} \\sqrt{2}"
} | {
"split": "train",
"index": 36944,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence 1, 2, 1, 2, 2, 1, 2, 2, 2, 1..., where each pair of 1s is separated by 2s, and there are n 2s between the nth pair of 1s, the sum of the first 1234 terms of this sequence is ______. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "2419"
} | {
"split": "train",
"index": 36945,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A,$ $B,$ $C$ be the angles of a non-right triangle. Compute\n\\[\\begin{vmatrix} \\tan A & 1 & 1 \\\\ 1 & \\tan B & 1 \\\\ 1 & 1 & \\tan C \\end{vmatrix}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 36946,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In convex pentagon $ABCDE$, angles $A$, $B$ and $C$ are congruent and angles $D$ and $E$ are congruent. If the measure of angle $A$ is 40 degrees less than the measure of angle $D$, what is the measure of angle $D$? Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "132^\\circ"
} | {
"split": "train",
"index": 36947,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of inches in the perimeter of a square is equal to the number of square inches in its area. Find the length, in inches, of a side of the square. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 36948,
"solution": ""
} |
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