data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Suppose that $x$ is real number such that $\\frac{27\\times 9^x}{4^x}=\\frac{3^x}{8^x}$ . Find the value of $2^{-(1+\\log_23)x}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "216"
} | {
"split": "train",
"index": 26349,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest natural number that is a multiple of 80, in which two of its distinct digits can be rearranged so that the resulting number is also a multiple of 80. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1520"
} | {
"split": "train",
"index": 26350,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Gauss is a famous German mathematician, known as the \"Prince of Mathematics\". There are 110 achievements named after \"Gauss\". Let $x\\in\\mathbb{R}$, use $[x]$ to represent the largest integer not exceeding $x$, and use $\\{x\\}=x-[x]$ to represent the non-negative fraction... | math | {
"style": "rule",
"ground_truth": "3024+ \\sqrt {3}"
} | {
"split": "train",
"index": 26351,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square carpet of side length 9 feet is designed with one large shaded square and eight smaller, congruent shaded squares, as shown. [asy]\n\ndraw((0,0)--(9,0)--(9,9)--(0,9)--(0,0));\n\nfill((1,1)--(2,1)--(2,2)--(1,2)--cycle,gray(.8));\n\nfill((4,1)--(5,1)--(5,2)--(4,2)--cycle... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 26352,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the squares of the solutions to\n\\[\\left| x^2 - x + \\frac{1}{2023} \\right| = \\frac{1}{2023}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4042}{2023}"
} | {
"split": "train",
"index": 26353,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the $1992^{\\text{nd}}$ letter in this sequence?\n\\[\\text{ABCDEDCBAABCDEDCBAABCDEDCBAABCDEDC}\\cdots\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "C"
} | {
"split": "train",
"index": 26354,
"solution": "1. **Identify the pattern and length of the repeating sequence**: The sequence given is $\\text{ABCDEDCBA}$, which repeats indefinitely. The length of this sequence is 9 letters.\n\n2. **Determine the position within the repeating sequence**: To find the $1992^\\te... |
deepscaler | [
{
"role": "user",
"content": "Let $A B C$ be an isosceles triangle with apex $A$. Let $I$ be the incenter. If $A I=3$ and the distance from $I$ to $B C$ is 2 , then what is the length of $B C$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{5}"
} | {
"split": "train",
"index": 26355,
"solution": "Let $X$ and $Y$ be the points where the incircle touches $A B$ and $B C$, respectively. Then $A X I$ and $A Y B$ are similar right triangles. Since $I$ is the incenter, we have $I X=I Y=2$. Using the Pythagorean theorem on triangle $A X I$, we find $A X=\\sqrt{5}$.... |
deepscaler | [
{
"role": "user",
"content": "Determine the value of the following sum:\n\n$$\n\\log _{3}\\left(1-\\frac{1}{15}\\right)+\\log _{3}\\left(1-\\frac{1}{14}\\right)+\\log _{3}\\left(1-\\frac{1}{13}\\right)+\\cdots+\\log _{3}\\left(1-\\frac{1}{8}\\right)+\\log _{3}\\left(1-\\frac{1}{7}\\right)+\\log _{3}\\left(1... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 26356,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the monotonic intervals of $F(x)=\\int_{0}^{x}(t^{2}+2t-8)dt$ for $x > 0$.\n\n(1) Determine the monotonic intervals of $F(x)$.\n\n(2) Find the maximum and minimum values of the function $F(x)$ on the interval $[1,3]$. Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "-\\frac{28}{3}"
} | {
"split": "train",
"index": 26357,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the system of equations in terms of $x$ and $y$: $\\left\\{\\begin{array}{l}{3x+5y=m+2}\\\\{2x+3y=m}\\end{array}\\right.$.<br/>$(1)$ Find the relationship between $x$ and $y$ (express $y$ as an algebraic expression only containing $x$);<br/>$(2)$ If the solution to this s... | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"split": "train",
"index": 26358,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rectangle $ABCD$ has sides $\\overline {AB}$ of length 4 and $\\overline {CB}$ of length 3. Divide $\\overline {AB}$ into 168 congruent segments with points $A=P_0, P_1, \\ldots, P_{168}=B$, and divide $\\overline {CB}$ into 168 congruent segments with points $C=Q_0, Q_1, \\ldo... | math | {
"style": "rule",
"ground_truth": "840"
} | {
"split": "train",
"index": 26359,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( n \\) be a natural number. Define \\( 1 = d_{1} < d_{2} < d_{3} < \\cdots < d_{k} = n \\) as its divisors. It is noted that \\( n = d_{2}^{2} + d_{3}^{3} \\). Determine all possible values of \\( n \\). Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "68"
} | {
"split": "train",
"index": 26360,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct arrangements of the letters in the word \"balloon\" are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1260"
} | {
"split": "train",
"index": 26361,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. Given that $b^{2}=ac$ and $a^{2}-c^{2}=ac-bc$.\n1. Find the measure of $\\angle A$.\n2. Let $f(x)=\\cos (wx-\\frac{A}{2})+\\sin (wx) (w > 0)$ and the smallest positive period of $f(x... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 26362,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The minimum possible sum of the three dimensions of a rectangular box with a volume of 3003 in^3 is what value? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 26363,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a, b, c$ are real numbers such that $a+b+c=6$ and $ab+bc+ca = 9$ , find the sum of all possible values of the expression $\\lfloor a \\rfloor + \\lfloor b \\rfloor + \\lfloor c \\rfloor$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 26364,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a debate competition with 4 participants, the rules are as follows: each participant must choose one topic from two options, A and B. For topic A, answering correctly earns 100 points, and answering incorrectly results in a loss of 100 points. For topic B, answering correctl... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 26365,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right cylinder with a height of 5 inches has a radius of 3 inches. A cone with a height of 3 inches and the same radius is placed on top of the cylinder. What is the total surface area of the combined object, in square inches? Express your answer in terms of \\(\\pi\\). Let's... | math | {
"style": "rule",
"ground_truth": "30\\pi + 9\\sqrt{2}\\pi"
} | {
"split": "train",
"index": 26366,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x,$ $y,$ $z$ be real numbers such that $-1 < x,$ $y,$ $z < 1.$ Find the minimum value of\n\\[\\frac{1}{(1 - x)(1 - y)(1 - z)} + \\frac{1}{(1 + x)(1 + y)(1 + z)}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 26367,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer with exactly 12 positive integer divisors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "288"
} | {
"split": "train",
"index": 26368,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $n$ pawns on $n$ distinct squares of a $19\\times 19$ chessboard. In each move, all the pawns are simultaneously moved to a neighboring square (horizontally or vertically) so that no two are moved onto the same square. No pawn can be moved along the same line in... | math | {
"style": "rule",
"ground_truth": "361"
} | {
"split": "train",
"index": 26369,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a>0$, $b>1$, and $a+b=2$, find the minimum value of $$\\frac{1}{2a}+\\frac{2}{b-1}$$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"split": "train",
"index": 26370,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( \\triangle ABC \\), if \\( \\overrightarrow{AB} \\cdot \\overrightarrow{AC} = 7 \\) and \\( |\\overrightarrow{AB} - \\overrightarrow{AC}| = 6 \\), find the maximum possible area of \\( \\triangle ABC \\). Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 26371,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a$, $b$, $c \\in \\mathbb{R}$, $a + b + c = 3$, $a \\geqslant b \\geqslant c$, the equation $ax^{2}+bx+c=0$ has real roots. Find the minimum value of $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 26372,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each face of a fair six-sided die is marked with one of the numbers $1, 2, \\cdots, 6$. When two such identical dice are rolled, the sum of the numbers on the top faces of these dice is the score for that roll. What is the probability that the product of the scores from three s... | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"split": "train",
"index": 26373,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "I randomly pick an integer $p$ between $1$ and $20$ inclusive. What is the probability that I choose a $p$ such that there exists an integer $q$ so that $p$ and $q$ satisfy the equation $pq - 6p - 3q = 3$? Express your answer as a common fraction. Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{20}"
} | {
"split": "train",
"index": 26374,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that there are $m$ distinct positive even numbers and $n$ distinct positive odd numbers such that their sum is 2015. Find the maximum value of $20m + 15n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1105"
} | {
"split": "train",
"index": 26375,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the functions $f(x)=x-\\frac{1}{x}$ and $g(x)=2a\\ln x$.\n\n(1) When $a\\geqslant -1$, find the monotonically increasing interval of $F(x)=f(x)-g(x)$;\n\n(2) Let $h(x)=f(x)+g(x)$, and $h(x)$ has two extreme values $({{x}_{1}},{{x}_{2}})$, where ${{x}_{1}}\\in (0,\\frac{1}... | math | {
"style": "rule",
"ground_truth": "\\frac{20\\ln 3-16}{3}"
} | {
"split": "train",
"index": 26376,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the expansion of $(1+x)^3 + (1+x)^4 + \\ldots + (1+x)^{12}$, the coefficient of the term containing $x^2$ is _____. (Provide your answer as a number) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "285"
} | {
"split": "train",
"index": 26377,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $15$ and $25$ meters. What fraction of the yard is occupied by the flow... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"split": "train",
"index": 26378,
"solution": "1. **Identify the dimensions of the triangles:**\n The problem states that the yard contains two congruent isosceles right triangles and the parallel sides of the trapezoidal remainder are $15$ meters and $25$ meters. The difference in length between the parallel... |
deepscaler | [
{
"role": "user",
"content": "Each of the numbers 1, 2, 3, and 4 is substituted, in some order, for \\( p, q, r \\), and \\( s \\). Find the greatest possible value of \\( p^q + r^s \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "83"
} | {
"split": "train",
"index": 26379,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When simplified, $(-\frac{1}{125})^{-2/3}$ becomes: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 26380,
"solution": "1. **Rewrite the expression with positive exponent**: The expression $(-\\frac{1}{125})^{-2/3}$ can be rewritten by taking the reciprocal of the base to make the exponent positive:\n \\[\n (-\\frac{1}{125})^{-2/3} = (-125)^{2/3}\n \\]\n\n2. **Evaluate the exp... |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the digits of the base $7$ representation of $2019_{10}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 26381,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(The full score of this question is 14 points) It is known that A and B are two fixed points on a plane, and the moving point P satisfies $|PA| + |PB| = 2$.\n(1) Find the equation of the trajectory of point P;\n(2) Suppose the line $l: y = k (k > 0)$ intersects the trajectory o... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 26382,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the $xy$ -coordinate plane, the $x$ -axis and the line $y=x$ are mirrors. If you shoot a laser beam from the point $(126, 21)$ toward a point on the positive $x$ -axis, there are $3$ places you can aim at where the beam will bounce off the mirrors and eventually re... | math | {
"style": "rule",
"ground_truth": "111"
} | {
"split": "train",
"index": 26383,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express eleven in base 2. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1011_2"
} | {
"split": "train",
"index": 26384,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Vera has several identical matches, from which she makes a triangle. Vera wants any two sides of this triangle to differ in length by at least $10$ matches, but it turned out that it is impossible to add such a triangle from the available matches (it is impossible to leave ex... | math | {
"style": "rule",
"ground_truth": "62"
} | {
"split": "train",
"index": 26385,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sets\n$$\n\\begin{array}{c}\nM=\\{x, xy, \\lg (xy)\\} \\\\\nN=\\{0, |x|, y\\},\n\\end{array}\n$$\nand that \\( M = N \\), determine the value of\n$$\n\\left(x+\\frac{1}{y}\\right)+\\left(x^2+\\frac{1}{y^2}\\right)+\\left(x^3+\\frac{1}{y^3}\\right)+\\cdots+\\left(x^{20... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 26386,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$(100^2-99^2) + (98^2-97^2) + \\ldots + (2^2-1^2) = \\ $ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5050"
} | {
"split": "train",
"index": 26387,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The decimal number $13^{101}$ is given. It is instead written as a ternary number. What are the two last digits of this ternary number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 26388,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are two pairs $(x,y)$ of real numbers that satisfy the equation $x+y = 3xy = 4$. Given that the solutions $x$ are in the form $x = \\frac{a \\pm b\\sqrt{c}}{d}$ where $a$, $b$, $c$, and $d$ are positive integers and the expression is completely simplified, what is the val... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 26389,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the random variables \\( X \\sim N(1,2) \\) and \\( Y \\sim N(3,4) \\), if \\( P(X < 0) = P(Y > a) \\), find the value of \\( a \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 + \\sqrt{2}"
} | {
"split": "train",
"index": 26390,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parabola C: y^2 = 4x, point B(3,0), and the focus F, point A lies on C. If |AF| = |BF|, calculate the length of |AB|. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 26391,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( a, b, c \\geq 0 \\), \\( t \\geq 1 \\), and satisfying\n\\[\n\\begin{cases}\na + b + c = \\frac{1}{2}, \\\\\n\\sqrt{a + \\frac{1}{2}(b - c)^{2}} + \\sqrt{b} + \\sqrt{c} = \\frac{\\sqrt{6t}}{2},\n\\end{cases}\n\\]\n\nfind \\( a^{2t} + b^{2t} + c^{2t} \\). Let's think s... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"split": "train",
"index": 26392,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fair die, numbered 1, 2, 3, 4, 5, 6, is thrown three times. The numbers obtained are recorded sequentially as $a$, $b$, and $c$. The probability that $a+bi$ (where $i$ is the imaginary unit) is a root of the equation $x^{2}-2x+c=0$ is $\\_\\_\\_\\_\\_\\_$. Let's think step by... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{108}"
} | {
"split": "train",
"index": 26393,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $T_n$ be the sum of the reciprocals of the non-zero digits of the integers from $1$ to $5^n$ inclusive. Find the smallest positive integer $n$ for which $T_n$ is an integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 26394,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( a > 0 \\). The function \\( f(x) = x + \\frac{100}{x} \\) attains its minimum value \\( m_{1} \\) on the interval \\( (0, a] \\), and its minimum value \\( m_{2} \\) on the interval \\( [a, +\\infty) \\). If \\( m_{1} m_{2} = 2020 \\), find the value of \\( a \\). Let's... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 26395,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a cube with a fly standing at each of its vertices. When a whistle blows, each fly moves to a vertex in the same face as the previous one but diagonally opposite to it. After the whistle blows, in how many ways can the flies change position so that there is no vertex w... | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 26396,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, it is known that $\\cos C + (\\cos A - \\sqrt{3} \\sin A) \\cos B = 0$.\n\n(1) Find the measure of angle $B$.\n\n(2) If $\\sin (A - \\frac{\\pi}{3}) = \\frac{3}{5}$, find $\\sin 2C$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac{24 + 7\\sqrt{3}}{50}"
} | {
"split": "train",
"index": 26397,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The pattern of Pascal's triangle is illustrated in the diagram shown. What is the fourth element in Row 15 of Pascal's triangle? $$\n\\begin{array}{ccccccccccccc}\\vspace{0.1in}\n\\textrm{Row 0}: & \\qquad & & & & & 1 & & & & & & \\\\ \\vspace{0.1in}\n\\textrm{Row 1}: & \\qqua... | math | {
"style": "rule",
"ground_truth": "455"
} | {
"split": "train",
"index": 26398,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cylinder is filled with gas at atmospheric pressure (103.3 kPa). Assuming the gas is ideal, determine the work (in joules) during the isothermal compression of the gas by a piston that has moved inside the cylinder by $h$ meters.\n\nHint: The equation of state for the gas is ... | math | {
"style": "rule",
"ground_truth": "900"
} | {
"split": "train",
"index": 26399,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An Ultraman is fighting a group of monsters. It is known that Ultraman has one head and two legs. Initially, each monster has two heads and five legs. During the battle, some monsters split, with each splitting monster creating two new monsters, each with one head and six legs ... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 26400,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $\\frac{2013^3-2 \\cdot 2013^2 \\cdot 2014+3 \\cdot 2013 \\cdot 2014^2-2014^3+1}{2013 \\cdot 2014}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2013"
} | {
"split": "train",
"index": 26401,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\cos 90^\\circ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 26402,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $PQR$, $\\cos(2P-Q) + \\sin(P+Q) = 2$ and $PQ = 5$. What is $QR$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5\\sqrt{3}"
} | {
"split": "train",
"index": 26403,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$f(x)$ is a monic polynomial such that $f(0)=4$ and $f(1)=10$. If $f(x)$ has degree $2$, what is $f(x)$? Express your answer in the form $ax^2+bx+c$, where $a$, $b$, and $c$ are real numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x^2+5x+4"
} | {
"split": "train",
"index": 26404,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\dbinom{1293}{1}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1293"
} | {
"split": "train",
"index": 26405,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Abe, Bobby, Charles, Devin and Edwin are the participants in a race. How many different 1st-2nd-3rd place outcomes are possible if there are no ties? Two different outcomes to include are Bobby-Devin-Edwin and Devin-Bobby-Edwin. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 26406,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "While Travis is having fun on cubes, Sherry is hopping in the same manner on an octahedron. An octahedron has six vertices and eight regular triangular faces. After five minutes, how likely is Sherry to be one edge away from where she started? Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{16}"
} | {
"split": "train",
"index": 26407,
"solution": "Let the starting vertex be the 'bottom' one. Then there is a 'top' vertex, and 4 'middle' ones. If $p(n)$ is the probability that Sherry is on a middle vertex after $n$ minutes, $p(0)=0$, $p(n+1)=(1-p(n))+p(n) \\cdot \\frac{1}{2}$. This recurrence gives us the foll... |
deepscaler | [
{
"role": "user",
"content": "Let the number $9999\\cdots 99$ be denoted by $N$ with $94$ nines. Then find the sum of the digits in the product $N\\times 4444\\cdots 44$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "846"
} | {
"split": "train",
"index": 26408,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ellipse whose equation is\n\\[\\frac{x^2}{25} + \\frac{y^2}{9} = 1\\]is graphed below. The chord $\\overline{AB}$ passes through a focus $F$ of the ellipse. If $AF = \\frac{3}{2},$ then find $BF.$\n\n[asy]\nunitsize (0.6 cm);\n\npair A, B, F;\n\nF = (4,0);\nA = (35/8,3*sq... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{4}"
} | {
"split": "train",
"index": 26409,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $m=(\\sqrt{3}\\sin \\omega x,\\cos \\omega x)$, $n=(\\cos \\omega x,-\\cos \\omega x)$ ($\\omega > 0$, $x\\in\\mathbb{R}$), $f(x)=m\\cdot n-\\frac{1}{2}$ and the distance between two adjacent axes of symmetry on the graph of $f(x)$ is $\\frac{\\pi}{2}$.\n\n$(1)$ Find the ... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{4}"
} | {
"split": "train",
"index": 26410,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $XYZ$, medians $XM$ and $YN$ intersect at $Q$, $QN=3$, $QM=4$, and $MN=5$. What is the area of $XMYN$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 26411,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest prime $p$ such that $p$ divides $2^{p+1} + 3^{p+1} + 5^{p+1} + 7^{p+1}$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 26412,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $3.57 - 1.14 - 0.23$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2.20"
} | {
"split": "train",
"index": 26413,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( a, b, c, d, e \\) be natural numbers with \\( a < b < c < d < e \\), and \\( a + 2b + 3c + 4d + 5e = 300 \\). Determine the maximum value of \\( a + b \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 26414,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A journey is divided into three segments: uphill, flat, and downhill. The ratio of the lengths of these segments is $1: 2: 3$, and the ratio of the times taken to walk each segment is $4: 5: 6$. Given that the person’s speed uphill is 3 km/h and the total distance is 50 km, how... | math | {
"style": "rule",
"ground_truth": "10.4166666667"
} | {
"split": "train",
"index": 26415,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the value of the expression $(\\square + 121 \\times 3.125) \\div 121$ is approximately 3.38, what natural number should be placed in $\\square$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 26416,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "As shown in the diagram, three circles intersect to create seven regions. Fill the integers $0 \\sim 6$ into the seven regions such that the sum of the four numbers within each circle is the same. What is the maximum possible value of this sum? Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 26417,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest value among $\\operatorname{lcm}[12,2],$ $\\operatorname{lcm}[12,4],$ $\\operatorname{lcm}[12,6],$ $\\operatorname{lcm}[12,8],$ $\\operatorname{lcm}[12,10],$ and $\\operatorname{lcm}[12,12]?$ Express your answer as an integer. Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 26418,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The maximum value of the function \\( y = \\frac{\\sin x \\cos x}{1 + \\sin x + \\cos x} \\) is $\\quad$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2} - 1}{2}"
} | {
"split": "train",
"index": 26419,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all real numbers $x$ that are not in the domain of the function $$g(x) = \\frac{1}{2 + \\frac{1}{2 + \\frac{1}{x}}}.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{9}{10}"
} | {
"split": "train",
"index": 26420,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the equation $\\frac{n!}{2}=k!+l!$ in natural numbers $n$, $k$, and $l$, where $n! = 1 \\cdot 2 \\cdots n$. In your answer, indicate 0 if there are no solutions, specify the value of $n$ if there is only one solution, or provide the sum of all $n$ values if there are mult... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 26421,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a polygon drawn on graph paper with a perimeter of 2014 units, and whose sides follow the grid lines, what is the maximum possible area of this polygon? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "253512"
} | {
"split": "train",
"index": 26422,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\log_{10} (x^2-3x+6)=1$, the value of $x$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4 or -1"
} | {
"split": "train",
"index": 26423,
"solution": "1. **Convert the logarithmic equation to an exponential equation:**\n Given the equation $\\log_{10} (x^2-3x+6)=1$, we can rewrite it using the definition of logarithms:\n \\[\n x^2 - 3x + 6 = 10^1\n \\]\n Simplifying the right-hand side, we have:\n \\[... |
deepscaler | [
{
"role": "user",
"content": "Find the volume of a cylinder formed by rotating a square with side length 10 centimeters about its horizontal line of symmetry. Express your answer in terms of $\\pi$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "250\\pi"
} | {
"split": "train",
"index": 26424,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A \\(101 \\times 101\\) grid is given, where all cells are initially colored white. You are allowed to choose several rows and paint all the cells in those rows black. Then, choose exactly the same number of columns and invert the color of all cells in those columns (i.e., chan... | math | {
"style": "rule",
"ground_truth": "5100"
} | {
"split": "train",
"index": 26425,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The cookies in a cookie jar contain a total of 100 raisins. All but one of the cookies are the same size and contain the same number of raisins. One cookie is larger and contains one more raisin than each of the others. The number of cookies in the jar is between 5 and 10, incl... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 26426,
"solution": "Let $n$ be the number of cookies in the cookie jar. Let $r$ be the number of raisins in each of the $n-1$ smaller, identical cookies. This means that there are $r+1$ raisins in the larger cookie. If we removed one raisin from the larger cookie, it too would have $r... |
deepscaler | [
{
"role": "user",
"content": "Let $L(m)$ be the $x$ coordinate of the left end point of the intersection of the graphs of $y=x^2-6$ and $y=m$, where $-6<m<6$. Let $r=[L(-m)-L(m)]/m$. Then, as $m$ is made arbitrarily close to zero, the value of $r$ is: Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{\\sqrt{6}}"
} | {
"split": "train",
"index": 26427,
"solution": "1. **Identify the intersection points**: The intersection points of the graphs of $y = x^2 - 6$ and $y = m$ are given by solving the equation:\n \\[ x^2 - 6 = m \\]\n \\[ x^2 = m + 6 \\]\n \\[ x = \\pm \\sqrt{m + 6} \\]\n Thus, the $x$-coordinates of the in... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the side lengths are: $AB = 17, BC = 20$ and $CA = 21$. $M$ is the midpoint of side $AB$. The incircle of $\\triangle ABC$ touches $BC$ at point $D$. Calculate the length of segment $MD$.\nA) $7.5$\nB) $8.5$\nC) $\\sqrt{8.75}$\nD) $9.5$ Let's think step by ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{8.75}"
} | {
"split": "train",
"index": 26428,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $\\frac{x^{2}}{4}-y^{2}=1$ with its right focus $F$, and points $P_{1}$, $P_{2}$, …, $P_{n}$ on its right upper part where $2\\leqslant x\\leqslant 2 \\sqrt {5}, y\\geqslant 0$. The length of the line segment $|P_{k}F|$ is $a_{k}$, $(k=1,2,3,…,n)$. If the se... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 26429,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\begin{cases} (\\frac{1}{2})^{x} & x\\geqslant 4 \\\\ f(x+1) & x < 4 \\end{cases}$, find the value of $f(2+\\log_{2}3)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{24}"
} | {
"split": "train",
"index": 26430,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the product of $1101_2 \\cdot 111_2$. Express your answer in base 2. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10010111_2"
} | {
"split": "train",
"index": 26431,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 10 mountaineers, divided equally into two groups. Among them, 4 are familiar with the trails. Each group needs 2 people who are familiar with the trails. The number of different ways to distribute them is: Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 26432,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(1) Simplify: $\\dfrac{\\tan(3\\pi-\\alpha)\\cos(2\\pi-\\alpha)\\sin(-\\alpha+\\dfrac{3\\pi}{2})}{\\cos(-\\alpha-\\pi)\\sin(-\\pi+\\alpha)\\cos(\\alpha+\\dfrac{5\\pi}{2})}$;\n\n(2) Given $\\tan\\alpha=\\dfrac{1}{4}$, find the value of $\\dfrac{1}{2\\cos^{2}\\alpha-3\\sin\\alpha... | math | {
"style": "rule",
"ground_truth": "\\dfrac{17}{20}"
} | {
"split": "train",
"index": 26433,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $\\dfrac{{x}^{2}}{16}+ \\dfrac{{y}^{2}}{9}=1$, with left and right foci $F_1$ and $F_2$ respectively, and a point $P$ on the ellipse, if $P$, $F_1$, and $F_2$ are the three vertices of a right triangle, calculate the distance from point $P$ to the $x$-axis. Le... | math | {
"style": "rule",
"ground_truth": "\\dfrac{9}{4}"
} | {
"split": "train",
"index": 26434,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Petya invented four distinct natural numbers and wrote down all their pairwise sums on the board. Below those, he wrote all their sums taken three at a time. It turned out that the sum of the two largest pairwise sums and the two smallest sums from those taken three at a time (... | math | {
"style": "rule",
"ground_truth": "1006"
} | {
"split": "train",
"index": 26435,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all pairs of integers $ (x,y)$, such that\n\\[ x^2 \\minus{} 2009y \\plus{} 2y^2 \\equal{} 0\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(0,0)； (-588,784)； (588,784)"
} | {
"split": "train",
"index": 26436,
"solution": "\nTo solve the equation \\(x^2 - 2009y + 2y^2 = 0\\) for integer pairs \\((x, y)\\), we begin by rearranging the equation as follows:\n\n\\[\nx^2 = 2009y - 2y^2.\n\\]\n\nThe right-hand side must be a perfect square for some integer \\(x\\). Therefore, consider the ... |
deepscaler | [
{
"role": "user",
"content": "Given that the area of $\\triangle ABC$ is $\\frac{1}{2}$, $AB=1$, $BC=\\sqrt{2}$, determine the value of $AC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 26437,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCDEF$ be a regular hexagon. Points $P$ and $Q$ on tangents to its circumcircle at $A$ and $D$ respectively are such that $PQ$ touches the minor arc $EF$ of this circle. Find the angle between $PB$ and $QC$ . Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 26438,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a triangle \\( ABC \\) where \\( |AB| = |AC| \\) and \\( \\angle BAC = 80^\\circ \\). Inside the triangle, there is a point \\( M \\) such that \\( \\angle MBC = 30^\\circ \\) and \\( \\angle MCB = 10^\\circ \\). Find \\( \\angle AMC \\). Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 26439,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Harold, Tanya, and Ulysses paint a very long picket fence.\nHarold starts with the first picket and paints every $h$ th picket;\nTanya starts with the second picket and paints every $t$ th picket; and\nUlysses starts with the third picket and paints every $u$ th picket.\n Call ... | math | {
"style": "rule",
"ground_truth": "757"
} | {
"split": "train",
"index": 26440,
"solution": "The three conditions state that no picket number $n$ may satisfy any two of the conditions: $n \\equiv 1 \\pmod{h},\\ n \\equiv 2 \\pmod{t},\\ n \\equiv 3 \\pmod{u}$. Note that the smallest number, $min \\{ h,t,u \\},$ divides the other $2$, and the next smallest d... |
deepscaler | [
{
"role": "user",
"content": "The expression $12y^2-65y+42$ can be written as $(Ay-14)(By-3),$ where $A$ and $B$ are integers. What is $AB + A$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 26441,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rationalize the denominator: $$\\frac{1}{\\sqrt[3]{2}+\\sqrt[3]{16}}$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt[3]{4}}{6}"
} | {
"split": "train",
"index": 26442,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $1$. The polygons meet at a point $A$ in such a way that the sum of the three interior angles at $A$ is $360^{\\circ}$. Thus the three polygons form a new polygon with $A... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 26443,
"solution": "1. **Identify the Problem Requirements:**\n We need to find three non-overlapping regular polygons, at least two of which are congruent, that meet at a point $A$ such that the sum of their interior angles at $A$ is $360^\\circ$. The sides of each polygon are of l... |
deepscaler | [
{
"role": "user",
"content": "What is the base $2$ representation of $125_{10}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1111101_2"
} | {
"split": "train",
"index": 26444,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are many ways in which the list \\(0,1,2,3,4,5,6,7,8,9\\) can be separated into groups. For example, this list could be separated into the four groups \\(\\{0,3,4,8\\}\\), \\(\\{1,2,7\\}\\), \\{6\\}, and \\{5,9\\}. The sum of the numbers in each of these four groups is \\... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 26445,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle DEF$ with sides $5$, $12$, and $13$, a circle with center $Q$ and radius $2$ rolls around inside the triangle, always keeping tangency to at least one side of the triangle. When $Q$ first returns to its original position, through what distance has $Q$ traveled? L... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 26446,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A basketball team has 18 players, including two sets of twins: Ben & Jerry and Tom & Tim. In how many ways can we choose 5 starters if at least one member from each set of twins must be included in the starting lineup? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "1834"
} | {
"split": "train",
"index": 26447,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that there are 4 tiers of horses, with Tian Ji's top-tier horse being better than King Qi's middle-tier horse, worse than King Qi's top-tier horse, Tian Ji's middle-tier horse being better than King Qi's bottom-tier horse, worse than King Qi's middle-tier horse, and Tian ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 26448,
"solution": ""
} |
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