data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Tim plans a weeklong prank to repeatedly steal Nathan's fork during lunch. He involves different people each day:\n- On Monday, he convinces Joe to do it.\n- On Tuesday, either Betty or John could undertake the prank.\n- On Wednesday, there are only three friends from whom he c... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 34149,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(ABCD\\) be an isosceles trapezoid with \\(AB=1, BC=DA=5, CD=7\\). Let \\(P\\) be the intersection of diagonals \\(AC\\) and \\(BD\\), and let \\(Q\\) be the foot of the altitude from \\(D\\) to \\(BC\\). Let \\(PQ\\) intersect \\(AB\\) at \\(R\\). Compute \\(\\sin \\angl... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"split": "train",
"index": 34150,
"solution": "Let \\(M\\) be the foot of the altitude from \\(B\\) to \\(CD\\). Then \\(2CM+AB=CD \\Longrightarrow CM=3\\). Then \\(DM=4\\) and by the Pythagorean theorem, \\(BM=4\\). Thus \\(BMD\\) is a right isosceles triangle i.e. \\(\\angle BDM=\\angle PDC=\\frac{\\pi}{4}\\)... |
deepscaler | [
{
"role": "user",
"content": "Given \\(1990 = 2^{\\alpha_{1}} + 2^{\\alpha_{2}} + \\cdots + 2^{\\alpha_{n}}\\), where \\(\\alpha_{1}, \\alpha_{2}, \\cdots, \\alpha_{n}\\) are distinct non-negative integers. Find \\(\\alpha_{1} + \\alpha_{2} + \\cdots + \\alpha_{n}\\). Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "43"
} | {
"split": "train",
"index": 34151,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the geometric sequence $\\{a_n\\}$, $a_3a_4a_5 = 3$ and $a_6a_7a_8 = 24$, calculate the value of $a_9a_{10}a_{11}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "192"
} | {
"split": "train",
"index": 34152,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $f(x)$ is an odd function defined on $\\mathbb{R}$ and satisfies $f(x+2)=- \\frac{1}{f(x)}$, and $f(x)=x$ when $1 \\leq x \\leq 2$, find $f(- \\frac{11}{2})$. 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": 34153,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four people, A, B, C, and D, participated in an exam. The combined scores of A and B are 17 points higher than the combined scores of C and D. A scored 4 points less than B, and C scored 5 points more than D. How many points higher is the highest score compared to the lowest sc... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 34154,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $\\{a_{n}\\}$ satisfies $a_{1}+\\frac{1}{2}{a_2}+\\frac{1}{3}{a_3}+…+\\frac{1}{n}{a_n}={a_{n+1}}-1$, $n\\in N^{*}$, and $a_{1}=1$.<br/>$(1)$ Find the general formula for the sequence $\\{a_{n}\\}$;<br/>$(2)$ Let $S_{n}=a_{1}\\cdot a_{n}+a_{2}\\cdot a_{n-1}+a_{3}\\c... | math | {
"style": "rule",
"ground_truth": "1012"
} | {
"split": "train",
"index": 34155,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define the polynomials $P_0, P_1, P_2 \\cdots$ by:\n\\[ P_0(x)=x^3+213x^2-67x-2000 \\]\n\\[ P_n(x)=P_{n-1}(x-n), n \\in N \\]\nFind the coefficient of $x$ in $P_{21}(x)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "61610"
} | {
"split": "train",
"index": 34156,
"solution": "\nTo find the coefficient of \\( x \\) in \\( P_{21}(x) \\), we need to evaluate the transformation of the polynomial \\( P_0(x) \\) through a series of substitutions as defined by the recurrence relation \\( P_n(x) = P_{n-1}(x-n) \\).\n\nInitially, we have:\n\\[ P... |
deepscaler | [
{
"role": "user",
"content": "Let $N$ be the number of positive integers that are less than or equal to $2003$ and whose base-$2$ representation has more $1$'s than $0$'s. Find the remainder when $N$ is divided by $1000$.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "155"
} | {
"split": "train",
"index": 34157,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $|a|=3$, $|b-2|=9$, and $a+b > 0$, find the value of $ab$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-33"
} | {
"split": "train",
"index": 34158,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a set of points \\( M \\) on a plane and seven different circles \\( C_{1}, C_{2}, \\dots, C_{7} \\). Circle \\( C_{7} \\) passes through exactly 7 points in \\( M \\); circle \\( C_{6} \\) passes through exactly 6 points in \\( M \\); ..., circle \\( C_{1} \\) passes ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 34159,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a quadrilateral with an inscribed circle, centre $O$ . Let \n\\[AO = 5, BO =6, CO = 7, DO = 8.\\] \nIf $M$ and $N$ are the midpoints of the diagonals $AC$ and $BD$ , determine $\\frac{OM}{ON}$ . Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "35/48"
} | {
"split": "train",
"index": 34160,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all the prime numbers between 1 and 10? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 34161,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a natural number \\( a \\), let \\( S(a) \\) represent the sum of its digits (for example, \\( S(123) = 1 + 2 + 3 = 6 \\) ). If a natural number \\( n \\) has all distinct digits, and \\( S(3n) = 3S(n) \\), what is the maximum value of \\( n \\)? Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "3210"
} | {
"split": "train",
"index": 34162,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( x_{1}, x_{2}, \\cdots, x_{n} \\) and \\( a_{1}, a_{2}, \\cdots, a_{n} \\) be two sets of arbitrary real numbers (where \\( n \\geqslant 2 \\)) that satisfy the following conditions:\n1. \\( x_{1} + x_{2} + \\cdots + x_{n} = 0 \\)\n2. \\( \\left| x_{1} \\right| + \\left|... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 34163,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\sin 225^\\circ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 34164,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four lighthouses are located at points $A$, $B$, $C$, and $D$. The lighthouse at $A$ is $5$ kilometers from the lighthouse at $B$, the lighthouse at $B$ is $12$ kilometers from the lighthouse at $C$, and the lighthouse at $A$ is $13$ kilometers from the lighthouse at $C$. To an... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 34165,
"solution": "Let $O$ be the intersection of $BC$ and $AD$. By the Angle Bisector Theorem, $\\frac {5}{BO}$ = $\\frac {13}{CO}$, so $BO$ = $5x$ and $CO$ = $13x$, and $BO$ + $OC$ = $BC$ = $12$, so $x$ = $\\frac {2}{3}$, and $OC$ = $\\frac {26}{3}$. Let $P$ be the foot of the alti... |
deepscaler | [
{
"role": "user",
"content": "Let \\(p\\), \\(q\\), \\(r\\), and \\(s\\) be the roots of the polynomial \\[x^4 + 10x^3 + 20x^2 + 15x + 6 = 0.\\] Find the value of \\[\\frac{1}{pq} + \\frac{1}{pr} + \\frac{1}{ps} + \\frac{1}{qr} + \\frac{1}{qs} + \\frac{1}{rs}.\\] Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "\\frac{10}{3}"
} | {
"split": "train",
"index": 34166,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given positive integers \\(a\\), \\(b\\) \\((a \\leq b)\\), a sequence \\(\\{ f_{n} \\}\\) satisfies:\n\\[ f_{1} = a, \\, f_{2} = b, \\, f_{n+2} = f_{n+1} + f_{n} \\text{ for } n = 1, 2, \\ldots \\]\n\nIf for any positive integer \\(n\\), it holds that \n\\[ \\left( \\sum_{k=1}... | math | {
"style": "rule",
"ground_truth": "2 + \\sqrt{5}"
} | {
"split": "train",
"index": 34167,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Reduce the number $\\sqrt[3]{2+\\sqrt{5}}+\\sqrt[3]{2-\\sqrt{5}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 34168,
"solution": "Observe that $(\\sqrt[3]{2+\\sqrt{5}}+\\sqrt[3]{2-\\sqrt{5}})^{3}=(2+\\sqrt{5})-3(\\sqrt[3]{2+\\sqrt{5}})-3(\\sqrt[3]{2-\\sqrt{5}})+(2-\\sqrt{5})=4-3(\\sqrt[3]{2+\\sqrt{5}}+\\sqrt[3]{2-\\sqrt{5}})$ Hence $\\sqrt[3]{2+\\sqrt{5}}+\\sqrt[3]{2-\\sqrt{5}}$ is a root of ... |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of $3^{4}$ points in four-dimensional space where each coordinate is in $\\{-1,0,1\\}$. Let $N$ be the number of sequences of points $P_{1}, P_{2}, \\ldots, P_{2020}$ in $S$ such that $P_{i} P_{i+1}=2$ for all $1 \\leq i \\leq 2020$ and $P_{1}=(0,0,0,0)$. (He... | math | {
"style": "rule",
"ground_truth": "4041"
} | {
"split": "train",
"index": 34169,
"solution": "From $(0,0,0,0)$ we have to go to $( \\pm 1, \\pm 1, \\pm 1, \\pm 1)$, and from $(1,1,1,1)$ (or any of the other similar points), we have to go to $(0,0,0,0)$ or $(-1,1,1,1)$ and its cyclic shifts. If $a_{i}$ is the number of ways to go from $(1,1,1,1)$ to point of... |
deepscaler | [
{
"role": "user",
"content": "Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides 7 times as fast as he walks, and both choices require the same amount of time. What is the... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 34170,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When a positive integer $x$ is divided by a positive integer $y$, the quotient is $u$ and the remainder is $v$, where $u$ and $v$ are integers.\nWhat is the remainder when $x+2uy$ is divided by $y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "v"
} | {
"split": "train",
"index": 34171,
"solution": "1. **Understanding the division algorithm**: According to the division algorithm, when a positive integer $x$ is divided by another positive integer $y$, we can express $x$ as:\n \\[\n x = uy + v\n \\]\n where $u$ is the quotient and $v$ is the remainder. H... |
deepscaler | [
{
"role": "user",
"content": "William is biking from his home to his school and back, using the same route. When he travels to school, there is an initial $20^\\circ$ incline for $0.5$ kilometers, a flat area for $2$ kilometers, and a $20^\\circ$ decline for $1$ kilometer. If William travels at $... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 34172,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive divisors does 6! have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 34173,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When programming a computer to print the first 10,000 natural numbers greater than 0: $1,2,3, \\cdots, 10000$, the printer unfortunately has a malfunction. Each time it prints the digits 7 or 9, it prints $x$ instead. How many numbers are printed incorrectly? Let's think step b... | math | {
"style": "rule",
"ground_truth": "5904"
} | {
"split": "train",
"index": 34174,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, $\\triangle ABE$, $\\triangle BCE$ and $\\triangle CDE$ are right-angled triangles with $\\angle AEB=\\angle BEC = \\angle CED = 45^\\circ$ and $AE=32$. Find the length of $CE.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 34175,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\tan x = -\\frac{12}{5}$ and $x \\in (\\frac{\\pi}{2}, \\pi)$, find the value of $\\cos(-x + \\frac{3\\pi}{2})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{12}{13}"
} | {
"split": "train",
"index": 34176,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Estimate the population of the island of Thalassa in the year 2050, knowing that its population doubles every 20 years and increases by an additional 500 people every decade thereafter, given that the population in the year 2000 was 250. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "1500"
} | {
"split": "train",
"index": 34177,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Eighty percent of adults drink coffee and seventy percent drink tea. What is the smallest possible percent of adults who drink both coffee and tea? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50\\%"
} | {
"split": "train",
"index": 34178,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 7 people standing in a row. How many different arrangements are there according to the following requirements?\n(1) Among them, A, B, and C cannot stand next to each other;\n(2) Among them, A and B have exactly one person between them;\n(3) A does not stand at the hea... | math | {
"style": "rule",
"ground_truth": "3720"
} | {
"split": "train",
"index": 34179,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute: $55\\times1212-15\\times1212$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48480"
} | {
"split": "train",
"index": 34180,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $m$ is an integer and $0 < 3m < 27$, what is the sum of all possible integer values of $m$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 34181,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the lengths of the arcs of the curves given by the equations in the rectangular coordinate system.\n\n$$\ny = e^{x} + e, \\ln \\sqrt{3} \\leq x \\leq \\ln \\sqrt{15}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 + \\frac{1}{2} \\ln \\left( \\frac{9}{5} \\right)"
} | {
"split": "train",
"index": 34182,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The first AMC $8$ was given in $1985$ and it has been given annually since that time. Samantha turned $12$ years old the year that she took the seventh AMC $8$. In what year was Samantha born? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1979"
} | {
"split": "train",
"index": 34183,
"solution": "1. **Identify the year of the seventh AMC 8**: \n The first AMC 8 was given in 1985. Since the AMC 8 is held annually, the subsequent AMC 8s would be in the years 1986, 1987, 1988, 1989, 1990, and the seventh AMC 8 would be in 1991.\n\n2. **Determine Samantha's a... |
deepscaler | [
{
"role": "user",
"content": "The digits 1, 2, 3, 4, and 5 were used, each one only once, to write a certain five-digit number \\(abcde\\) such that \\(abc\\) is divisible by 4, \\(bcd\\) is divisible by 5, and \\(cde\\) is divisible by 3. Find this number. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "12453"
} | {
"split": "train",
"index": 34184,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the base edge length of a regular square pyramid is $2$, and the side edge length is $\\sqrt{6}$, determine the volume of the pyramid. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{3}"
} | {
"split": "train",
"index": 34185,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a set $S$ consisting of two undefined elements \"pib\" and \"maa\", and the four postulates: $P_1$: Every pib is a collection of maas, $P_2$: Any two distinct pibs have one and only one maa in common, $P_3$: Every maa belongs to two and only two pibs, $P_4$: There are exa... | math | {
"style": "rule",
"ground_truth": "all"
} | {
"split": "train",
"index": 34186,
"solution": "To solve this problem, we need to verify if each of the theorems $T_1$, $T_2$, and $T_3$ can be deduced from the given postulates $P_1$, $P_2$, $P_3$, and $P_4$.\n\n#### Step 1: Verify Theorem $T_1$ (There are exactly six maas)\nFrom $P_4$, we know there are exactl... |
deepscaler | [
{
"role": "user",
"content": "Let \\[f(x) =\n\\begin{cases}\n2x^2 - 3&\\text{if } x\\le 2, \\\\\nax + 4 &\\text{if } x>2.\n\\end{cases}\n\\]Find $a$ if the graph of $y=f(x)$ is continuous (which means the graph can be drawn without lifting your pencil from the paper). Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 34187,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are three positive integers: large, medium, and small. The sum of the large and medium numbers equals 2003, and the difference between the medium and small numbers equals 1000. What is the sum of these three positive integers? Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "2004"
} | {
"split": "train",
"index": 34188,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a} = (1, -1)$ and $\\overrightarrow{b} = (6, -4)$, if $\\overrightarrow{a}$ is perpendicular to $(t\\overrightarrow{a} + \\overrightarrow{b})$, find the value of the real number $t$. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 34189,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that Liliane has $30\\%$ more cookies than Jasmine and Oliver has $10\\%$ less cookies than Jasmine, and the total number of cookies in the group is $120$, calculate the percentage by which Liliane has more cookies than Oliver. Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "44.44\\%"
} | {
"split": "train",
"index": 34190,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $$a_{n}= \\frac {n(n+1)}{2}$$, remove all the numbers in the sequence $\\{a_n\\}$ that can be divided by 2, and arrange the remaining numbers in ascending order to form the sequence $\\{b_n\\}$. Find the value of $b_{21}$. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "861"
} | {
"split": "train",
"index": 34191,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a positive constant $c,$ in spherical coordinates $(\\rho,\\theta,\\phi),$ find the shape described by the equation\n\\[\\rho = c.\\](A) Line\n(B) Circle\n(C) Plane\n(D) Sphere\n(E) Cylinder\n(F) Cone\n\nEnter the letter of the correct option. Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "\\text{(D)}"
} | {
"split": "train",
"index": 34192,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle BAC$, $\\angle BAC=40^\\circ$, $AB=10$, and $AC=6$. Points $D$ and $E$ lie on $\\overline{AB}$ and $\\overline{AC}$ respectively. What is the minimum possible value of $BE+DE+CD$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 34193,
"solution": "1. **Reflect Points Across Lines**: Reflect point $C$ across line $AB$ to get point $C'$ and reflect point $B$ across line $AC$ to get point $B'$. This reflection ensures that $BE = B'E$ and $CD = C'D$ due to the properties of reflection.\n\n2. **Collinearity of Po... |
deepscaler | [
{
"role": "user",
"content": "For $x$ real, the inequality $1 \\le |x-2| \\le 7$ is equivalent to Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-5 \\le x \\le 1$ or $3 \\le x \\le 9"
} | {
"split": "train",
"index": 34194,
"solution": "To solve the inequality $1 \\leq |x-2| \\leq 7$, we need to consider the absolute value inequality in two parts: the lower bound and the upper bound.\n\n1. **Solving the lower bound inequality $|x-2| \\geq 1$:**\n - The absolute value inequality $|x-2| \\geq 1$ i... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\alpha$ is an angle in the second quadrant, and $P(x, 4)$ is a point on its terminal side, with $\\cos\\alpha= \\frac {1}{5}x$, then $x= \\_\\_\\_\\_\\_\\_$, and $\\tan\\alpha= \\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "-\\frac {4}{3}"
} | {
"split": "train",
"index": 34195,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three of the following test scores are Cyprian's and the other three are Margaret's: 85, 87, 92, 93, 94, 98. Cyprian's mean score is 90. What is Margaret's mean score? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "93"
} | {
"split": "train",
"index": 34196,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nCompute the definite integral:\n\n$$\n\\int_{0}^{\\pi} 2^{4} \\cdot \\sin ^{6}\\left(\\frac{x}{2}\\right) \\cos ^{2}\\left(\\frac{x}{2}\\right) d x\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{8}"
} | {
"split": "train",
"index": 34197,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $C$: $\\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, passing through point $Q(\\sqrt{2}, 1)$ and having the right focus at $F(\\sqrt{2}, 0)$,\n(I) Find the equation of the ellipse $C$;\n(II) Let line $l$: $y = k(x - 1) (k > 0)$ intersect the $x$... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{42}}{2}"
} | {
"split": "train",
"index": 34198,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $a_{2012}$ if $a_{n} \\equiv a_{n-1}+n(\\bmod 2012)$ and $a_{1}=1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1006"
} | {
"split": "train",
"index": 34199,
"solution": "Taking both sides modulo 2012, we see that $a_{n} \\equiv a_{n-1}+n(\\bmod 2012)$. Therefore, $a_{2012} \\equiv a_{2011}+2012 \\equiv a_{2010}+2011+2012 \\equiv \\ldots \\equiv 1+2+\\ldots+2012 \\equiv \\frac{(2012)(2013)}{2} \\equiv(1006)(2013) \\equiv (1006)(1) \... |
deepscaler | [
{
"role": "user",
"content": "Rectangle $EFGH$ has an area of $4032$. An ellipse with area $4032\\pi$ passes through points $E$ and $G$ and has foci at $F$ and $H$. Determine the perimeter of rectangle $EFGH$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8\\sqrt{2016}"
} | {
"split": "train",
"index": 34200,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Bob rolls a fair six-sided die each morning. If Bob rolls a composite number, he eats sweetened cereal. If he rolls a prime number, he eats unsweetened cereal. If he rolls a 1, then he rolls again. In a non-leap year, what is the expected value of the difference between the num... | math | {
"style": "rule",
"ground_truth": "73"
} | {
"split": "train",
"index": 34201,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\sin 2α - 2 = 2\\cos 2α$, find the value of $\\sin^{2}α + \\sin 2α$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{5}"
} | {
"split": "train",
"index": 34202,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the line $x-y+2=0$ and the circle $C$: $(x-3)^{2}+(y-3)^{2}=4$ intersect at points $A$ and $B$. The diameter through the midpoint of chord $AB$ is $MN$. Calculate the area of quadrilateral $AMBN$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{2}"
} | {
"split": "train",
"index": 34203,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a new configuration, six circles with a radius of 5 units intersect at a single point. What is the number of square units in the area of the shaded region? The region is formed similarly to the original problem where the intersections create smaller sector-like areas. Expres... | math | {
"style": "rule",
"ground_truth": "75\\pi - 25\\sqrt{3}"
} | {
"split": "train",
"index": 34204,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In convex quadrilateral $KLMN$ side $\\overline{MN}$ is perpendicular to diagonal $\\overline{KM}$, side $\\overline{KL}$ is perpendicular to diagonal $\\overline{LN}$, $MN = 65$, and $KL = 28$. The line through $L$ perpendicular to side $\\overline{KN}$ intersects diagonal $\\... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 34205,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Isabella uses one-foot cubical blocks to build a rectangular fort that is $12$ feet long, $10$ feet wide, and $5$ feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain? Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "280"
} | {
"split": "train",
"index": 34206,
"solution": "1. **Calculate the volume of the original rectangular fort**: \n The fort is built using the dimensions given as $12$ feet in length, $10$ feet in width, and $5$ feet in height. The volume of a rectangular prism is calculated by multiplying its length, width, and... |
deepscaler | [
{
"role": "user",
"content": "The Bulls are playing the Knicks in the NBA playoffs. To win this playoff series, a team must secure 4 victories before the other team. If the Knicks win each game with a probability of $\\dfrac{3}{5}$ and there are no ties, what is the probability that the Bulls will win the p... | math | {
"style": "rule",
"ground_truth": "\\frac{864}{15625}"
} | {
"split": "train",
"index": 34207,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_{1}, a_{2}, a_{3}, \\ldots$ be a sequence of positive real numbers that satisfies $$\\sum_{n=k}^{\\infty}\\binom{n}{k} a_{n}=\\frac{1}{5^{k}}$$ for all positive integers $k$. The value of $a_{1}-a_{2}+a_{3}-a_{4}+\\cdots$ can be expressed as $\\frac{a}{b}$, where $a, b$ ... | math | {
"style": "rule",
"ground_truth": "542"
} | {
"split": "train",
"index": 34208,
"solution": "Let $S_{k}=\\frac{1}{5^{k}}$. In order to get the coefficient of $a_{2}$ to be -1, we need to have $S_{1}-3S_{3}$. This subtraction makes the coefficient of $a_{3}$ become -6. Therefore, we need to add $7S_{3}$ to make the coefficient of $a_{4}$ equal to 1. The coe... |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has vertices with coordinates $A(2,3),$ $B(7,8),$ and $C(-4,6)$. The triangle is reflected about line $L$. The image points are $A'(2,-5),$ $B'(7,-10),$ and $C'(-4,-8)$. What is the equation of line $L$? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "y = -1"
} | {
"split": "train",
"index": 34209,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the isosceles triangle \\(ABC\\) (\\(AB = BC\\)), a point \\(D\\) is taken on the side \\(BC\\) such that \\(BD : DC = 1 : 4\\). In what ratio does the line \\(AD\\) divide the altitude \\(BE\\) of triangle \\(ABC\\), counting from vertex \\(B\\)? Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "1:2"
} | {
"split": "train",
"index": 34210,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nAn aluminum cube with an edge length of \\( l = 10 \\) cm is heated to a temperature of \\( t_{1} = 100^{\\circ} \\mathrm{C} \\). After this, it is placed on ice, which has a temperature of \\( t_{2} = 0^{\\circ} \\mathrm{C} \\). Determine the maximum depth to which the cube ... | math | {
"style": "rule",
"ground_truth": "0.0818"
} | {
"split": "train",
"index": 34211,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It takes Clea 60 seconds to walk down an escalator when it is not moving, and 24 seconds when it is moving. How many seconds would it take Clea to ride the escalator down when she is not walking? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 34212,
"solution": "1. **Identify the variables and equations:**\n Let $x$ be Clea's walking speed in units per second and $y$ be the total distance of the escalator in units. When Clea walks down the stationary escalator, it takes her 60 seconds, so the equation is:\n \\[\n 60x... |
deepscaler | [
{
"role": "user",
"content": "Determine the integer $x$ that satisfies the following set of congruences:\n\\begin{align*}\n4+x &\\equiv 3^2 \\pmod{2^3} \\\\\n6+x &\\equiv 4^2 \\pmod{3^3} \\\\\n8+x &\\equiv 6^2 \\pmod{5^3}\n\\end{align*}\nFind the remainder when $x$ is divided by $120$. Let's think step by s... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 34213,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all ordered pairs $(m, n)$ of integers such that $231 m^{2}=130 n^{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(0,0)"
} | {
"split": "train",
"index": 34214,
"solution": "The unique solution is $(0,0)$."
} |
deepscaler | [
{
"role": "user",
"content": "In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of $\\alpha$, $\\beta$, and $\\alpha + \\beta$ radians, respectively, where $\\alpha + \\beta < \\pi$. If $\\cos \\alpha$, which is a positive rational number, is expressed as a fraction in lowest term... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 34215,
"solution": "Using the first diagram above, \\[\\sin \\frac{\\alpha}{2} = \\frac{1}{r}\\] \\[\\sin \\frac{\\beta}{2} = \\frac{1.5}{r}\\] \\[\\sin(\\frac{\\alpha}{2}+\\frac{\\beta}{2})=\\frac{2}{r}\\] by the Pythagorean trig identities, \\[\\cos\\frac{\\alpha}{2}=\\sqrt{1-\\frac... |
deepscaler | [
{
"role": "user",
"content": "Given that 216 sprinters enter a 100-meter dash competition, and the track has 6 lanes, determine the minimum number of races needed to find the champion sprinter. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "43"
} | {
"split": "train",
"index": 34216,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $d$ is a digit. For how many values of $d$ is $2.00d5 > 2.005$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 34217,
"solution": "To solve this problem, we need to compare the number $2.00d5$ with $2.005$ for each possible digit $d$ (where $d$ can be any digit from $0$ to $9$).\n\n1. **Expression of $2.00d5$:** \n The number $2.00d5$ can be expressed as $2.00d5 = 2 + 0.00d + 0.0005$. Here,... |
deepscaler | [
{
"role": "user",
"content": "On each side of an equilateral triangle, a point is taken. The sides of the triangle with vertices at these points are perpendicular to the sides of the original triangle. In what ratio does each of these points divide the side of the original triangle? Let's think step by step... | math | {
"style": "rule",
"ground_truth": "1:2"
} | {
"split": "train",
"index": 34218,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jack, Jill, and John play a game in which each randomly picks and then replaces a card from a standard 52 card deck, until a spades card is drawn. What is the probability that Jill draws the spade? (Jack, Jill, and John draw in that order, and the game repeats if no spade is dr... | math | {
"style": "rule",
"ground_truth": "\\frac{12}{37}"
} | {
"split": "train",
"index": 34219,
"solution": "The desired probability is the relative probability that Jill draws the spade. In the first round, Jack, Jill, and John draw a spade with probability $1 / 4,3 / 4 \\cdot 1 / 4$, and $(3 / 4)^{2} \\cdot 1 / 4$ respectively. Thus, the probability that Jill draws the ... |
deepscaler | [
{
"role": "user",
"content": "John is planning to fence a rectangular garden such that the area is at least 150 sq. ft. The length of the garden should be 20 ft longer than its width. Additionally, the total perimeter of the garden must not exceed 70 ft. What should the width, in feet, be? Let's think step ... | math | {
"style": "rule",
"ground_truth": "-10 + 5\\sqrt{10}"
} | {
"split": "train",
"index": 34220,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that Tian Ji's top horse is faster than the King of Qi's middle horse, and Tian Ji's middle horse is faster than the King of Qi's bottom horse, but Tian Ji's top horse is slower than the King of Qi's top horse, and Tian Ji's bottom horse is slower than the King of Qi's bo... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 34221,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A two-digit integer $AB$ equals $\\frac{1}{9}$ of the three-digit integer $CCB$, where $C$ and $B$ represent distinct digits from 1 to 9. What is the smallest possible value of the three-digit integer $CCB$? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "225"
} | {
"split": "train",
"index": 34222,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\tan{\\alpha} = 3$, calculate:\n\n1. $$\\frac{4\\sin{\\alpha} - \\cos{\\alpha}}{3\\sin{\\alpha} + 5\\cos{\\alpha}}$$\n2. $$\\sin{\\alpha} \\cdot \\cos{\\alpha}$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{10}"
} | {
"split": "train",
"index": 34223,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain kingdom, the king has decided to build 25 new towns on 13 uninhabited islands so that on each island there will be at least one town. Direct ferry connections will be established between any pair of new towns which are on different islands. Determine the least poss... | math | {
"style": "rule",
"ground_truth": "222"
} | {
"split": "train",
"index": 34224,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The least common multiple of two numbers is 3780, and the greatest common divisor is 18. Given that one of the numbers is 180, what is the other number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "378"
} | {
"split": "train",
"index": 34225,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A geometric sequence of positive integers has its first term as 5 and its fourth term as 480. What is the second term of the sequence? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 34226,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It takes Mina 90 seconds to walk down an escalator when it is not operating, and 30 seconds to walk down when it is operating. Additionally, it takes her 40 seconds to walk up another escalator when it is not operating, and only 15 seconds to walk up when it is operating. Calcu... | math | {
"style": "rule",
"ground_truth": "69"
} | {
"split": "train",
"index": 34227,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\\times 5$ square array of dots? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 34228,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different right-angled triangles exist, one of the legs of which is \\(\\sqrt{2016}\\), and the other leg and hypotenuse are expressed in natural numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 34229,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( \\triangle ABC \\), the sides opposite to the angles \\( \\angle A \\), \\( \\angle B \\), and \\( \\angle C \\) are denoted as \\( a \\), \\( b \\), and \\( c \\) respectively. If \\( b^{2}=a^{2}+c^{2}-ac \\), and \\( c-a \\) is equal to the height \\( h \\) fr... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 34230,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If five squares of a $3 \\times 3$ board initially colored white are chosen at random and blackened, what is the expected number of edges between two squares of the same color?\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{16}{3}"
} | {
"split": "train",
"index": 34231,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On September 10, 2005, the following numbers were drawn in the five-number lottery: 4, 16, 22, 48, 88. All five numbers are even, exactly four of them are divisible by 4, three by 8, and two by 16. In how many ways can five different numbers with these properties be selected fr... | math | {
"style": "rule",
"ground_truth": "15180"
} | {
"split": "train",
"index": 34232,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the number \\(2016 * * * * 02 * *\\), each of the 6 asterisks needs to be replaced with any of the digits \\(0, 2, 4, 5, 7, 9\\) (digits can repeat) so that the resulting 12-digit number is divisible by 15. How many ways can this be done? Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "5184"
} | {
"split": "train",
"index": 34233,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a one-way single-lane highway, cars travel at the same speed and maintain a safety distance such that for every 20 kilometers per hour or part thereof in speed, there is a distance of one car length between the back of one car and the front of the next. Each car is 5 meters ... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 34234,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A=3^{7}+\\binom{7}{2}3^{5}+\\binom{7}{4}3^{3}+\\binom{7}{6}3$, $B=\\binom{7}{1}3^{6}+\\binom{7}{3}3^{4}+\\binom{7}{5}3^{2}+1$. Find $A-B$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 34235,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The 2007 AMC 10 will be scored by awarding $6$ points for each correct response, $0$ points for each incorrect response, and $1.5$ points for each problem left unanswered. After looking over the $25$ problems, Sarah has decided to attempt the first $22$ and leave only the last ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 34236,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Read the following material and complete the corresponding tasks.\n\n平衡多项式 \n定义:对于一组多项式$x+a$,$x+b$,$x+c$,$x+d(a$,$b$,$c$,$d$是常数）,当其中两个多项式的乘积与另外两个多项式乘积的差是一个常数$p$时,称这样的四个多项式是一组平衡多项式,$p$的绝对值是这组平衡多项式的平衡因子. \n例如:对于多项式$x+1$,$x+2$,$x+5$,$x+6$,因为$(x+1)(x+6)-(x+2)(x+5)=(x^{2}+7x+6)-(x... | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 34237,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest integer that must divide the product of any $4$ consecutive integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 34238,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The digits $1,2,3,4,5,$ and $6$ can be arranged to form many different $6$-digit positive integers with six distinct digits. In how many such integers is the digit $1$ to the left of both the digits $2$ and $3$? Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 34239,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice and Bob play a game involving a circle whose circumference is divided by 12 equally-spaced points. The points are numbered clockwise, from 1 to 12. Both start on point 12. Alice moves clockwise and Bob, counterclockwise.\nIn a turn of the game, Alice moves 5 points clockw... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 34240,
"solution": "1. **Identify the movement of Alice and Bob**: \n - Alice moves 5 points clockwise.\n - Bob moves 9 points counterclockwise, which is equivalent to moving 3 points clockwise (since 9 counterclockwise on a 12-point circle is the same as 3 clockwise: $12 - 9 = 3$... |
deepscaler | [
{
"role": "user",
"content": "The point \\((p, q)\\) is on the line \\(y=\\frac{2}{5} x\\). Also, the area of the rectangle shown is 90. What is the value of \\(p\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 34241,
"solution": "Since \\((p, q)\\) lies on the line \\(y=\\frac{2}{5} x\\), then \\(q=\\frac{2}{5} p\\). The given rectangle has two sides on the axes, so has width \\(p\\) and height \\(q\\). Therefore, the area of the rectangle equals \\(p q=p \\cdot \\frac{2}{5} p=\\frac{2}{5} ... |
deepscaler | [
{
"role": "user",
"content": "What is the remainder when the integer equal to \\( QT^2 \\) is divided by 100, given that \\( QU = 9 \\sqrt{33} \\) and \\( UT = 40 \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 34242,
"solution": "Let \\( O \\) be the centre of the top face of the cylinder and let \\( r \\) be the radius of the cylinder. We need to determine the value of \\( QT^2 \\). Since \\( RS \\) is directly above \\( PQ \\), then \\( RP \\) is perpendicular to \\( PQ \\). This means th... |
deepscaler | [
{
"role": "user",
"content": "The equation $2^{333x-2} + 2^{111x+2} = 2^{222x+1} + 1$ has three real roots. Given that their sum is $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "113"
} | {
"split": "train",
"index": 34243,
"solution": "Let $y = 2^{111x}$. Then our equation reads $\\frac{1}{4}y^3 + 4y = 2y^2 + 1$ or $y^3 - 8y^2 + 16y - 4 = 0$. Thus, if this equation has roots $r_1, r_2$ and $r_3$, by Vieta's formulas we have $r_1\\cdot r_2\\cdot r_3 = 4$. Let the corresponding values of $x$ be $x_... |
deepscaler | [
{
"role": "user",
"content": "Rectangle $ABCD$ is given with $AB=63$ and $BC=448.$ Points $E$ and $F$ lie on $AD$ and $BC$ respectively, such that $AE=CF=84.$ The inscribed circle of triangle $BEF$ is tangent to $EF$ at point $P,$ and the inscribed circle of triangle $DEF$ is tangent to $EF$ at point $Q.$ F... | math | {
"style": "rule",
"ground_truth": "259"
} | {
"split": "train",
"index": 34244,
"solution": "Several Pythagorean triples exist amongst the numbers given. $BE = DF = \\sqrt{63^2 + 84^2} = 21\\sqrt{3^2 + 4^2} = 105$. Also, the length of $EF = \\sqrt{63^2 + (448 - 2\\cdot84)^2} = 7\\sqrt{9^2 + 40^2} = 287$.\n Use the Two Tangent Theorem on $\\triangle BEF$. S... |
deepscaler | [
{
"role": "user",
"content": "Compute the sum of the geometric series $-1 + 2 - 4 + 8 - \\cdots + 512$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "341"
} | {
"split": "train",
"index": 34245,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many roots does the equation\n\n$$\n\\overbrace{f(f(\\ldots f}^{10 \\text{ times } f}(x) \\ldots)) + \\frac{1}{2} = 0\n$$\n\nhave, where \\( f(x) = |x| - 1 \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 34246,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all positive integers $n$ such that $\\sqrt{n^2+85n+2017}$ is an integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "195"
} | {
"split": "train",
"index": 34247,
"solution": "Notice that $(n+42)^2= n^2+84n+1764$. Also note that $(n+45)^2= n^2+90n+2025$. Thus, \\[(n+42)^2< n^2+85n+2017<(n+45)^2\\] where $n^2+85n+2017$ is a perfect square. Hence,\\[n^2+85n+2017= (n+43)^2\\] or \\[n^2+85n+2017= (n+44)^2.\\] Solving the two equations yields... |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all positive integers $n$ such that when $1^3+2^3+3^3+\\cdots +n^3$ is divided by $n+5$, the remainder is $17$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "239"
} | {
"split": "train",
"index": 34248,
"solution": "The formula for the sum of cubes is as follows: \\[1^3+2^3+3^3+...+n^3=(1+2+3+...+n)^2=\\left(\\frac{n(n+1)}{2}\\right)^2\\] So let's apply this to this problem.\n Let $m=n+5$. Then we have \\[1^3+2^3+3^3+\\cdots+(m-5)^3\\equiv 17 \\mod m\\] \\[\\left(\\frac{(m-5)(... |
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