data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"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 $AE$ . Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"split": "train",
"index": 33949,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 3 math teams in the area, with 5, 7, and 8 students respectively. Each team has two co-captains. If I randomly select a team, and then randomly select two members of that team to give a copy of $\\emph{Introduction to Geometry}$, what is the probability that both of t... | math | {
"style": "rule",
"ground_truth": "\\dfrac{11}{180}"
} | {
"split": "train",
"index": 33950,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Yan is somewhere between his office and a concert hall. To get to the concert hall, he can either walk directly there, or walk to his office and then take a scooter to the concert hall. He rides 5 times as fast as he walks, and both choices take the same amount of time. What is... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 33951,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfy $|\\overrightarrow{a}|=2$, $\\overrightarrow{b}=(4\\cos \\alpha,-4\\sin \\alpha)$, and $\\overrightarrow{a}\\perp (\\overrightarrow{a}- \\overrightarrow{b})$, let the angle between $\\overrightarrow{a}$ and $... | math | {
"style": "rule",
"ground_truth": "\\dfrac {\\pi}{3}"
} | {
"split": "train",
"index": 33952,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The integer $n$ has exactly six positive divisors, and they are: $1<a<b<c<d<n$ . Let $k=a-1$ . If the $k$ -th divisor (according to above ordering) of $n$ is equal to $(1+a+b)b$ , find the highest possible value of $n$ . Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "2009"
} | {
"split": "train",
"index": 33953,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$. For example, $R(15) = 1+0+3+0+3+1+7+6+5=26$. How many two-digit positive integers $n$ satisfy $R(n) = R(n+1)\\,?$ Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 33954,
"solution": "We start by defining the function $R(n)$ as the sum of the remainders when $n$ is divided by each integer from $2$ to $10$. We are interested in finding two-digit integers $n$ such that $R(n) = R(n+1)$.\n\n#### Step 1: Define the Change in Remainder Function\nLet $... |
deepscaler | [
{
"role": "user",
"content": "Simplify\n\\[(1 + \\cot A - \\csc A)(1 + \\tan A + \\sec A).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 33955,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a trapezoid \\(ABCD\\) with bases \\(AD=12\\) and \\(BC=8\\), circles constructed on the sides \\(AB\\), \\(BC\\), and \\(CD\\) as diameters intersect at one point. The length of diagonal \\(AC\\) is 12. Find the length of \\(BD\\). Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 33956,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Natural numbers are written in sequence on the blackboard, skipping over any perfect squares. The sequence looks like this:\n$$\n2,3,5,6,7,8,10,11, \\cdots\n$$\nThe first number is 2, the fourth number is 6, the eighth number is 11, and so on. Following this pattern, what is th... | math | {
"style": "rule",
"ground_truth": "2036"
} | {
"split": "train",
"index": 33957,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that in the geometric sequence $\\{a_n\\}$ where all terms are positive, $a_1a_3=16$ and $a_3+a_4=24$, find the value of $a_5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 33958,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $w$ and $z$ are complex numbers such that $|w+z|=2$ and $|w^2+z^2|=10,$ find the smallest possible value of $|w^4+z^4|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "82"
} | {
"split": "train",
"index": 33959,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A dragon is tethered by a 25-foot golden rope to the base of a sorcerer's cylindrical tower whose radius is 10 feet. The rope is attached to the tower at ground level and to the dragon at a height of 7 feet. The dragon has pulled the rope taut, the end of the rope is 5 feet fro... | math | {
"style": "rule",
"ground_truth": "862"
} | {
"split": "train",
"index": 33960,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Regular decagon $P_1 P_2 \\dotsb P_{10}$ is drawn in the coordinate plane with $P_1$ at $(1,0)$ and $P_6$ at $(3,0).$ If $P_n$ is the point $(x_n,y_n),$ compute the numerical value of the product\n\\[(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \\dotsm (x_{10} + y_{10} i).\\] Let's... | math | {
"style": "rule",
"ground_truth": "1023"
} | {
"split": "train",
"index": 33961,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine with proof the number of positive integers $n$ such that a convex regular polygon with $n$ sides has interior angles whose measures, in degrees, are integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 33962,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$, the sum of the first $n$ terms $S_n$ satisfies $a_{n+1}=2S_n+6$, and $a_1=6$.\n\n(Ⅰ) Find the general formula for the sequence $\\{a_n\\}$;\n\n(Ⅱ) Let $b_n=\\frac{a_n}{(a_n-2)(a_{n+1}-2)}$, and $T_n$ be the sum of the first $n$ terms of the sequenc... | math | {
"style": "rule",
"ground_truth": "m=1"
} | {
"split": "train",
"index": 33963,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of\n\\[\\sqrt{x^2 + (2 - x)^2} + \\sqrt{(2 - x)^2 + (2 + x)^2}\\]over all real numbers $x.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"split": "train",
"index": 33964,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a triangle, two angles measure 45 degrees and 60 degrees. The side opposite the 45-degree angle measures 8 units. Calculate the sum of the lengths of the other two sides. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19.3"
} | {
"split": "train",
"index": 33965,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The teams $T_1$, $T_2$, $T_3$, and $T_4$ are in the playoffs. In the semifinal matches, $T_1$ plays $T_4$, and $T_2$ plays $T_3$. The winners of those two matches will play each other in the final match to determine the champion. When $T_i$ plays $T_j$, the probability that $T_... | math | {
"style": "rule",
"ground_truth": "781"
} | {
"split": "train",
"index": 33966,
"solution": "There are two scenarios in which $T_4$ wins. The first scenario is where $T_4$ beats $T_1$, $T_3$ beats $T_2$, and $T_4$ beats $T_3$, and the second scenario is where $T_4$ beats $T_1$, $T_2$ beats $T_3$, and $T_4$ beats $T_2$. Consider the first scenario. The prob... |
deepscaler | [
{
"role": "user",
"content": "Given the expansion of the binomial $({x+\\frac{a}{{\\sqrt{x}}}})^n$ where $n\\in{N^*}$, in the expansion, ___, ___. Given the following conditions: ① the ratio of the binomial coefficients of the second term to the third term is $1:4$; ② the sum of all coefficients is $512$; ③... | math | {
"style": "rule",
"ground_truth": "-48"
} | {
"split": "train",
"index": 33967,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The letters $\\text{A}$, $\\text{J}$, $\\text{H}$, $\\text{S}$, $\\text{M}$, $\\text{E}$ and the digits $1$, $9$, $8$, $9$ are \"cycled\" separately as follows and put together in a numbered list:\n\\[\\begin{tabular}[t]{lccc} & & AJHSME & 1989 \\ & & & \\ 1. & & JHSMEA & 989... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 33968,
"solution": "1. **Identify the cycling pattern for letters and digits:**\n - The letters $\\text{AJHSME}$ are cycled by moving the first letter to the end. This cycle repeats every $6$ positions because there are $6$ letters.\n - The digits $1989$ are cycled by moving the f... |
deepscaler | [
{
"role": "user",
"content": "Let $x = 2001^{1002} - 2001^{-1002}$ and $y = 2001^{1002} + 2001^{-1002}.$ Find $x^2 - y^2.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 33969,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Vanya had a certain amount of cookies; he ate some, and then Tanya came to visit him and they divided the remaining cookies equally. It turned out that Vanya ate five times more cookies than Tanya did. What fraction of all the cookies did Vanya eat by the time Tanya arrived? Le... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{7}"
} | {
"split": "train",
"index": 33970,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Does there exist a three-digit number whose cube ends in three sevens? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "753"
} | {
"split": "train",
"index": 33971,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given set $A=\\{a-2, 12, 2a^2+5a\\}$, and $-3$ belongs to $A$, find the value of $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{2}"
} | {
"split": "train",
"index": 33972,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A vertex-induced subgraph is a subset of the vertices of a graph together with any edges whose endpoints are both in this subset. An undirected graph contains 10 nodes and $m$ edges, with no loops or multiple edges. What is the minimum possible value of $m$ such that this graph... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 33973,
"solution": "Suppose that we want to find the vertex-induced subgraph of maximum size where each vertex has degree at least 5. To do so, we start with the entire graph and repeatedly remove any vertex with degree less than 5. If there are vertices left after this process termin... |
deepscaler | [
{
"role": "user",
"content": "Alexio has 200 cards numbered 1-200, inclusive, and places them in a box. Alexio then chooses a card from the box at random. What is the probability that the number on the card he chooses is a multiple of 4, 5, or 7? Express your answer as a common fraction. Let's think step by... | math | {
"style": "rule",
"ground_truth": "\\frac{97}{200}"
} | {
"split": "train",
"index": 33974,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Katya is passing time while her parents are at work. On a piece of paper, she absentmindedly drew Cheburashkas in two rows (at least one Cheburashka was drawn in each row).\n\nAfterwards, she drew a Crocodile Gena between every two adjacent Cheburashkas in both rows. Then she d... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 33975,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\tan \\frac{\\pi}{24} + \\tan \\frac{7 \\pi}{24}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{6} - 2 \\sqrt{3}"
} | {
"split": "train",
"index": 33976,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 8 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{55}"
} | {
"split": "train",
"index": 33977,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\sigma_1 : \\mathbb{N} \\to \\mathbb{N}$ be a function that takes a natural number $n$ , and returns the sum of the positive integer divisors of $n$ . For example, $\\sigma_1(6) = 1 + 2 + 3 + 6 = 12$ . What is the largest number n such that $\\sigma_1(n) = 1854$ ?\n... | math | {
"style": "rule",
"ground_truth": "1234"
} | {
"split": "train",
"index": 33978,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f\\left(x\\right)=\\frac{2×202{3}^{x}}{202{3}^{x}+1}$, if the inequality $f(ae^{x})\\geqslant 2-f\\left(\\ln a-\\ln x\\right)$ always holds, then the minimum value of $a$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{e}"
} | {
"split": "train",
"index": 33979,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest $n$ such that $n$! ends in 290 zeroes. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1170"
} | {
"split": "train",
"index": 33980,
"solution": "Each 0 represents a factor of $10=2 \\cdot 5$. Thus, we wish to find the smallest factorial that contains at least 290 2's and 290 5's in its prime factorization. Let this number be $n$!, so the factorization of $n$! contains 2 to the power $p$ and 5 to the power $... |
deepscaler | [
{
"role": "user",
"content": "Given positive real numbers $a$, $b$, $c$, $d$ satisfying $a^{2}-ab+1=0$ and $c^{2}+d^{2}=1$, find the value of $ab$ when $\\left(a-c\\right)^{2}+\\left(b-d\\right)^{2}$ reaches its minimum. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2} + 1"
} | {
"split": "train",
"index": 33981,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, \\( PQR \\) is a straight line segment and \\( QS = QT \\). Also, \\( \\angle PQS = x^\\circ \\) and \\( \\angle TQR = 3x^\\circ \\). If \\( \\angle QTS = 76^\\circ \\), the value of \\( x \\) is: Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 33982,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 630 = 0$ has integral solutions? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "160"
} | {
"split": "train",
"index": 33983,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is known that each side and diagonal of a regular polygon is colored in one of exactly 2018 different colors, and not all sides and diagonals are the same color. If a regular polygon contains no two-colored triangles (i.e., a triangle whose three sides are precisely colored ... | math | {
"style": "rule",
"ground_truth": "2017^2"
} | {
"split": "train",
"index": 33984,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{A}$ and $\\mathbf{B}$ be matrices such that\n\\[\\mathbf{A} + \\mathbf{B} = \\mathbf{A} \\mathbf{B}.\\]If $\\mathbf{A} \\mathbf{B} = \\begin{pmatrix} 20/3 & 4/3 \\\\ -8/3 & 8/3 \\end{pmatrix},$ find $\\mathbf{B} \\mathbf{A}.$ Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 20/3 & 4/3 \\\\ -8/3 & 8/3 \\end{pmatrix}"
} | {
"split": "train",
"index": 33985,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbb{Z}$ be the set of all integers. Find all pairs of integers $(a,b)$ for which there exist functions $f \\colon \\mathbb{Z}\\rightarrow \\mathbb{Z}$ and $g \\colon \\mathbb{Z} \\rightarrow \\mathbb{Z}$ satisfying\n\\[ f(g(x))=x+a \\quad\\text{and}\\quad g(f(x))=x+b ... | math | {
"style": "rule",
"ground_truth": "\\left | a \\right | = \\left | b \\right |"
} | {
"split": "train",
"index": 33986,
"solution": "\nWe are tasked with finding all pairs of integers \\((a, b)\\) such that there exist functions \\( f: \\mathbb{Z} \\rightarrow \\mathbb{Z} \\) and \\( g: \\mathbb{Z} \\rightarrow \\mathbb{Z} \\) satisfying the conditions:\n\n\\[ f(g(x)) = x + a \\quad \\text{and} ... |
deepscaler | [
{
"role": "user",
"content": "What digit $B$ will make the number $527B$ divisible by $5$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0 \\text{ or } 5"
} | {
"split": "train",
"index": 33987,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Add $36_7 + 274_7.$ Express your answer in base 7. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "343_7"
} | {
"split": "train",
"index": 33988,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ is an equilateral triangle with each side of length 9. Points $D$, $E$, and $F$ are the trisection points of sides $AB$, $BC$, and $CA$ respectively, such that $D$ is closer to $A$, $E$ is closer to $B$, and $F$ is closer to $C$. Point $G$ is the midpoint of segm... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 33989,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\tan (3 \\alpha-2 \\beta)=\\frac{1}{2}$ and $\\tan (5 \\alpha-4 \\beta)=\\frac{1}{4}$, find $\\tan \\alpha$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{16}"
} | {
"split": "train",
"index": 33990,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In preparation for a game of Fish, Carl must deal 48 cards to 6 players. For each card that he deals, he runs through the entirety of the following process: 1. He gives a card to a random player. 2. A player Z is randomly chosen from the set of players who have at least as many... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"split": "train",
"index": 33991,
"solution": "After any number of cards are dealt, we see that the difference between the number of cards that any two players hold is at most one. Thus, after the first 47 cards have been dealt, there is only one possible distribution: there must be 5 players with 8 cards and 1... |
deepscaler | [
{
"role": "user",
"content": "For every real number $x$, let $[x]$ be the greatest integer which is less than or equal to $x$. If the postal rate for first class mail is six cents for every ounce or portion thereof, then the cost in cents of first-class postage on a letter weighing $W$ ounces is always Let'... | math | {
"style": "rule",
"ground_truth": "-6[-W]"
} | {
"split": "train",
"index": 33992,
"solution": "To solve this problem, we need to determine the cost of mailing a letter weighing $W$ ounces, where the rate is six cents for every ounce or portion thereof. This means that even a small fraction of an ounce over a whole number results in charging for an additional... |
deepscaler | [
{
"role": "user",
"content": "Find the smallest solution to the equation \\[\\frac{1}{x-2} + \\frac{1}{x-4} = \\frac{3}{x-3}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 - \\sqrt3"
} | {
"split": "train",
"index": 33993,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\dfrac{3+4i}{1+2i}$. Your answer should be of the form $a+bi$, where $a$ and $b$ are both real numbers and written as improper fractions (if necessary). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{11}{5} - \\dfrac{2}{5}i"
} | {
"split": "train",
"index": 33994,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify first, then evaluate: $(1-\\frac{2}{{m+1}})\\div \\frac{{{m^2}-2m+1}}{{{m^2}-m}}$, where $m=\\tan 60^{\\circ}-1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3-\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 33995,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\overrightarrow{a}$ and $\\overrightarrow{b}$ are non-zero vectors, $\\overrightarrow{m} = \\overrightarrow{a} + t \\overrightarrow{b} (t \\in \\mathbb{R})$, $|\\overrightarrow{a}| = 1$, $|\\overrightarrow{b}| = 2$, the minimum value of $|\\overrightarrow{m}|$ is o... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"split": "train",
"index": 33996,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Losyash is walking to Sovunya's house along the river at a speed of 4 km/h. Every half-hour, he launches paper boats that travel to Sovunya at a speed of 10 km/h. What is the time interval at which the boats arrive at Sovunya's house? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 33997,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ f(n)$ be a function defined on the set of all positive integers and having its values in the same set. Suppose that $ f(f(n) \\plus{} f(m)) \\equal{} m \\plus{} n$ for all positive integers $ n,m.$ Find the possible value for $ f(1988).$ Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "1988"
} | {
"split": "train",
"index": 33998,
"solution": "\nWe start with the functional equation provided:\n\n\\[\nf(f(n) + f(m)) = m + n\n\\]\n\nfor all positive integers \\( n \\) and \\( m \\).\n\nOur goal is to find the possible value for \\( f(1988) \\).\n\n1. **Substitute special values:**\n\n Let \\( n = m = 1 \... |
deepscaler | [
{
"role": "user",
"content": "Given a right triangle \\( ABC \\) with hypotenuse \\( AB \\). One leg \\( AC = 15 \\) and the altitude from \\( C \\) to \\( AB \\) divides \\( AB \\) into segments \\( AH \\) and \\( HB \\) with \\( HB = 16 \\). What is the area of triangle \\( ABC \\)? Let's think step by st... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 33999,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the remainder when $x^2+7x-5$ divides $2x^4+11x^3-42x^2-60x+47$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2x-8"
} | {
"split": "train",
"index": 34000,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Fill in the appropriate numbers in the parentheses:\n\n(1) 7÷9= $$\\frac {( )}{( )}$$\n\n(2) $$\\frac {12}{7}$$=\\_\\_\\_\\_÷\\_\\_\\_\\_\\_\n\n(3) 3 $$\\frac {5}{8}$$= $$\\frac {( )}{( )}$$\n\n(4) 6= $$\\frac {()}{11}$$ Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "\\frac {66}{11}"
} | {
"split": "train",
"index": 34001,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ $c,$ $d$ be real numbers such that\n\\[\\frac{(a - b)(c - d)}{(b - c)(d - a)} = \\frac{2}{5}.\\]Find the sum of all possible values of\n\\[\\frac{(a - c)(b - d)}{(a - b)(c - d)}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{2}"
} | {
"split": "train",
"index": 34002,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest two-digit prime number such that reversing the digits of the number produces a composite number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 34003,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the value of $b$, where $b$ is a positive number, such that the terms $10, b, \\frac{10}{9}, \\frac{10}{81}$ are the first four terms, respectively, of a geometric sequence. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 34004,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x)=\\sin (2017x+\\frac{\\pi }{6})+\\cos (2017x-\\frac{\\pi }{3})$, find the minimum value of $A|x_{1}-x_{2}|$ where $x_{1}$ and $x_{2}$ are real numbers such that $f(x_{1})\\leq f(x)\\leq f(x_{2})$ for all real $x$. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{2017}"
} | {
"split": "train",
"index": 34005,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circles with centers at $(5,5)$ and $(20,15)$ are both tangent to the $x$-axis. What is the distance between the closest points of the two circles? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5 \\sqrt{13} - 20"
} | {
"split": "train",
"index": 34006,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a WeChat group, there are five people playing the red envelope game: A, B, C, D, and E. There are 4 red envelopes, each person can grab at most one, and all red envelopes must be grabbed. Among the 4 red envelopes, there are two worth 2 yuan, one worth 3 yuan, and one worth ... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 34007,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Quarter circles of radius 1' form a pattern as shown below. What is the area, in square feet, of the shaded region in a 3-foot length of this pattern? The quarter circles' flat sides face outward, alternating top and bottom along the length. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}\\pi"
} | {
"split": "train",
"index": 34008,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the mass of the body $\\Omega$ with density $\\mu=z$, bounded by the surfaces\n\n$$\nx^{2} + y^{2} = 4, \\quad z=0, \\quad z=\\frac{x^{2} + y^{2}}{2}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{16\\pi}{3}"
} | {
"split": "train",
"index": 34009,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many values of $k$ is $60^{10}$ the least common multiple of the positive integers $10^{10}$, $12^{12}$, and $k$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "231"
} | {
"split": "train",
"index": 34010,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $N$ lockers, labeled from 1 to $N$, placed in clockwise order around a circular hallway. Initially, all lockers are open. Ansoon starts at the first locker and always moves clockwise. When she is at locker $n$ and there are more than $n$ open lockers, she keeps locker... | math | {
"style": "rule",
"ground_truth": "2046"
} | {
"split": "train",
"index": 34011,
"solution": "Note that in the first run-through, we will leave all lockers $2^{n}-1$ open. This is because after having locker $2^{n}-1$ open, we will close the next $2^{n}-1$ lockers and then start at locker $2^{n}-1+2^{n}-1+1=2^{n+1}-1$. Now we want 1 to be the last locker th... |
deepscaler | [
{
"role": "user",
"content": "Azar, Carl, Jon, and Sergey are the four players left in a singles tennis tournament. They are randomly assigned opponents in the semifinal matches, and the winners of those matches play each other in the final match to determine the winner of the tournament. When Azar plays Ca... | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 34012,
"solution": "Let $A$ be Azar, $C$ be Carl, $J$ be Jon, and $S$ be Sergey. The $4$ circles represent the $4$ players, and the arrow is from the winner to the loser with the winning probability as the label.\n This problem can be solved by using $2$ cases.\n$\\textbf{Case 1:}$ $C... |
deepscaler | [
{
"role": "user",
"content": "The absolute value of -1.2 is ____, and its reciprocal is ____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{5}{6}"
} | {
"split": "train",
"index": 34013,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Twenty tiles are numbered 1 through 20 and are placed into box $C$. Twenty other tiles numbered 15 through 34 are placed into box $D$. One tile is randomly drawn from each box. What is the probability that the tile from box $C$ is less than 18 and the tile from box $D$ is eithe... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{40}"
} | {
"split": "train",
"index": 34014,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $AB = 18$ and $BC = 12$. Find the largest possible value of $\\tan A$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{5}}{5}"
} | {
"split": "train",
"index": 34015,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of solutions to the equation\n\\[\\sin x = \\left( \\frac{1}{3} \\right)^x\\]\non the interval \\( (0, 150 \\pi) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 34016,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x,$ $y,$ and $z$ be three positive real numbers whose sum is 1. If $z = 2x$ and $y = 3x$, find the minimum value of the product $xyz.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{36}"
} | {
"split": "train",
"index": 34017,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a four-digit natural number $A$, if the digit in the thousands place is $5$ more than the digit in the tens place, and the digit in the hundreds place is $3$ more than the digit in the units place, then $A$ is called a \"five-three number.\" For example, for the four-digit ... | math | {
"style": "rule",
"ground_truth": "5401"
} | {
"split": "train",
"index": 34018,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The points \\( M \\) and \\( N \\) are chosen on the angle bisector \\( AL \\) of a triangle \\( ABC \\) such that \\( \\angle ABM = \\angle ACN = 23^\\circ \\). \\( X \\) is a point inside the triangle such that \\( BX = CX \\) and \\( \\angle BXC = 2\\angle BML \\). Find \\( ... | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 34019,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of the shape enclosed by the curve $y=x^2$ (where $x>0$), the tangent line at point A(2, 4), and the x-axis. 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": 34020,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A woman buys a property for $15,000, aiming for a $6\\%$ return on her investment annually. She sets aside $15\\%$ of the rent each month for maintenance, and pays $360 annually in taxes. What must be the monthly rent to meet her financial goals?\nA) $110.00$\nB) $123.53$\nC) $... | math | {
"style": "rule",
"ground_truth": "123.53"
} | {
"split": "train",
"index": 34021,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of ways to arrange the letters of the word SUCCESS. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "420"
} | {
"split": "train",
"index": 34022,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $(a,b,c,d)$ be an ordered quadruple of not necessarily distinct integers, each one of them in the set ${0,1,2,3}.$ For how many such quadruples is it true that $a \\cdot d-b \\cdot c$ is odd? (For example, $(0,3,1,1)$ is one such quadruple, because $0 \\cdot 1-3 \\cdot 1 = ... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 34023,
"solution": "\nTo solve the problem, we need to count the number of ordered quadruples $(a, b, c, d)$ such that $a\\cdot d - b\\cdot c$ is odd. We will use parity analysis to determine the conditions under which the expression is odd.\n\n#### Step 1: Understanding Parity of Pro... |
deepscaler | [
{
"role": "user",
"content": "Given that the terminal side of angle $\\alpha$ passes through the point $P(\\sqrt{3}, m)$ ($m \\neq 0$), and $\\cos\\alpha = \\frac{m}{6}$, then $\\sin\\alpha = \\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 34024,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\dbinom{505}{505}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 34025,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a \\(6 \\times 6\\) chessboard, we randomly place counters on three different squares. What is the probability that no two counters are in the same row or column? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40/119"
} | {
"split": "train",
"index": 34026,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an acute triangle $\\triangle ABC$, altitudes $\\overline{AD}$ and $\\overline{BE}$ intersect at point $H$. Given that $HD=6$ and $HE=3$, calculate $(BD)(DC)-(AE)(EC)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 34027,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the average of all the integer values of $M$ such that $\\frac{M}{56}$ is strictly between $\\frac{3}{7}$ and $\\frac{1}{4}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 34028,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A box contains twelve balls, each of a different color. Every minute, Randall randomly draws a ball from the box, notes its color, and then returns it to the box. Consider the following two conditions: (1) Some ball has been drawn at least twelve times (not necessarily consecut... | math | {
"style": "rule",
"ground_truth": "0.02236412255 \\ldots"
} | {
"split": "train",
"index": 34029,
"solution": "Below is a python implementation to compute the probability, using the same method as the solution to the easier version (with three balls). ``` from fractions import Fraction N = 12 probs = [{} for i in range ((N-1)*(N-1)+2)] prob1 = Fraction() prob2 = Fraction() ... |
deepscaler | [
{
"role": "user",
"content": "Six people stand in a row, with exactly two people between A and B. Calculate the number of different ways for them to stand. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 34030,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $x$ and $y$ are real numbers, and they satisfy $xy + x + y = 17$, $x^2y + xy^2 = 66$, find the value of $x^4 + x^3y + x^2y^2 + xy^3 + y^4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12499"
} | {
"split": "train",
"index": 34031,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\frac{1}{1+\\sqrt{3}} \\cdot \\frac{1}{1-\\sqrt{5}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{1 - \\sqrt{5} + \\sqrt{3} - \\sqrt{15}}"
} | {
"split": "train",
"index": 34032,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Inside the square $ABCD$, a point $M$ is taken such that $\\angle MAB = 60^{\\circ}$ and $\\angle MCD = 15^{\\circ}$. Find $\\angle MBC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 34033,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Lyla and Isabelle run on a circular track both starting at point \\( P \\). Lyla runs at a constant speed in the clockwise direction. Isabelle also runs in the clockwise direction at a constant speed 25% faster than Lyla. Lyla starts running first and Isabelle starts running wh... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 34034,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = x + \\sin(\\pi x) - 3$, study its symmetry center $(a, b)$ and find the value of $f\\left( \\frac {1}{2016} \\right) + f\\left( \\frac {2}{2016} \\right) + f\\left( \\frac {3}{2016} \\right) + \\ldots + f\\left( \\frac {4030}{2016} \\right) + f\\left(... | math | {
"style": "rule",
"ground_truth": "-8062"
} | {
"split": "train",
"index": 34035,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For integers $a$, $b$, $c$, and $d$, $(x^2+ax+b)(x^2+cx+d)=x^4+x^3-2x^2+17x-5$. What is the value of $a+b+c+d$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 34036,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the integral \\(\\int_{0}^{1} \\ln x \\ln (1-x) \\, dx\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 - \\frac{\\pi^2}{6}"
} | {
"split": "train",
"index": 34037,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that the roots of the polynomial $P(x)=x^3+ax^2+bx+c$ are $\\cos \\frac{2\\pi}7,\\cos \\frac{4\\pi}7,$ and $\\cos \\frac{6\\pi}7$, where angles are in radians. What is $abc$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{32}"
} | {
"split": "train",
"index": 34038,
"solution": "1. **Identify the roots and express them in terms of complex exponentials:**\n Let $z = e^{\\frac{2\\pi i}{7}}$. The roots of the polynomial $P(x) = x^3 + ax^2 + bx + c$ are given as $\\cos \\frac{2\\pi}{7}, \\cos \\frac{4\\pi}{7},$ and $\\cos \\frac{6\\pi}{7}$. ... |
deepscaler | [
{
"role": "user",
"content": "Figures $0$, $1$, $2$, and $3$ consist of $1$, $5$, $13$, and $25$ nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure 100? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "20201"
} | {
"split": "train",
"index": 34039,
"solution": "1. **Identify the pattern**: We observe that the number of unit squares in figures $0$, $1$, $2$, and $3$ are $1$, $5$, $13$, and $25$ respectively. We need to find a formula that describes this sequence.\n\n2. **Recognize the sequence type**: The sequence appears ... |
deepscaler | [
{
"role": "user",
"content": "The graph of a parabola has the following properties:\n\n$\\bullet$ It passes through the point $(1,5).$\n\n$\\bullet$ The $y$-coordinate of the focus is 3.\n\n$\\bullet$ Its axis of symmetry is parallel to the $x$-axis.\n\n$\\bullet$ Its vertex lies on the $y$-axis.\n\nExpress... | math | {
"style": "rule",
"ground_truth": "y^2 - 4x - 6y + 9 = 0"
} | {
"split": "train",
"index": 34040,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square piece of paper is modified by cutting out one smaller right triangle from each corner, each having one leg along a side of the square facing outside. The remaining form is a rectangle. If each triangle has legs of lengths $4$ units and $3$ units, and the original squar... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 34041,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The region shown is bounded by the arcs of circles having a radius 8 units, with each arc having a central angle measure of 45 degrees and intersecting at points of tangency. Determine the area of this region, which can be expressed in the form $a\\sqrt{b} + c\\pi$ square units... | math | {
"style": "rule",
"ground_truth": "-22"
} | {
"split": "train",
"index": 34042,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ be the product of any three consecutive positive odd integers. The largest integer dividing all such $P$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 34043,
"solution": "1. **Express the product $P$:** Let the three consecutive odd integers be $2n-1$, $2n+1$, and $2n+3$. Then the product $P$ is:\n \\[\n P = (2n-1)(2n+1)(2n+3)\n \\]\n\n2. **Simplify the expression for $P$:** We can expand this product:\n \\[\n P = (2n-1)(2... |
deepscaler | [
{
"role": "user",
"content": "In a right triangle where the ratio of the legs is 1:3, a perpendicular is dropped from the vertex of the right angle to the hypotenuse. Find the ratio of the segments created on the hypotenuse by this perpendicular. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "1:9"
} | {
"split": "train",
"index": 34044,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider two fictional states: Alpha and Beta. Alpha issues license plates with a format of two letters followed by four numbers, and then ending with one letter (LLNNNNL). Beta issues plates with three letters followed by three numbers and lastly one letter (LLLNNNL). Assume a... | math | {
"style": "rule",
"ground_truth": "281216000"
} | {
"split": "train",
"index": 34045,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Fill the numbers $1,2,\\cdots,36$ into a $6 \\times 6$ grid, placing one number in each cell, such that the numbers in each row are in increasing order from left to right. What is the minimum possible sum of the six numbers in the third column? Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 34046,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many four-digit whole numbers are there such that the leftmost digit is an odd prime, the second digit is a multiple of 3, and all four digits are different? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "616"
} | {
"split": "train",
"index": 34047,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of three numbers is $98$. The ratio of the first to the second is $\\frac {2}{3}$, \nand the ratio of the second to the third is $\\frac {5}{8}$. The second number is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 34048,
"solution": "1. Let the three numbers be $a$, $b$, and $c$. According to the problem, the sum of these numbers is given by:\n \\[ a + b + c = 98 \\]\n\n2. The ratio of the first number to the second number is $\\frac{2}{3}$, which can be expressed as:\n \\[ \\frac{a}{b} = \... |
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