data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Given a triangle $\\triangle ABC$ with area $S$ and sides $a$, $b$, $c$ that satisfy the equations: $S=a^{2}-(b-c)^{2}$, $b+c=8$, find the maximum value of the area $S$ of $\\triangle ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {64}{17}"
} | {
"split": "train",
"index": 28449,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The repeating decimal for $\\frac{5}{13}$ is $0.cdc\\ldots$ What is the value of the sum $c+d$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 28450,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different positive integers can be represented as a difference of two distinct members of the set $\\{1, 2, 3, 4, 5, 6 \\}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 28451,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a caravan with 100 camels, consisting of both one-humped and two-humped camels, with at least one of each kind. If you take any 62 camels, they will have at least half of the total number of humps in the caravan. Let \\( N \\) be the number of two-humped camels. How ma... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 28452,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The orthocenter of triangle $ABC$ divides altitude $\\overline{CF}$ into segments with lengths $HF = 6$ and $HC = 15.$ Calculate $\\tan A \\tan B.$\n\n[asy]\nunitsize (1 cm);\n\npair A, B, C, D, E, F, H;\n\nA = (0,0);\nB = (5,0);\nC = (4,4);\nD = (A + reflect(B,C)*(A))/2;\nE =... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{2}"
} | {
"split": "train",
"index": 28453,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "8 students are practicing for a math contest, and they divide into pairs to take a practice test. In how many ways can they be split up? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "105"
} | {
"split": "train",
"index": 28454,
"solution": "We create the pairs one at a time. The first person has 7 possible partners. Set this pair aside. Of the remaining six people, pick a person. He or she has 5 possible partners. Set this pair aside. Of the remaining four people, pick a person. He or she has 3 possib... |
deepscaler | [
{
"role": "user",
"content": "A sequence of integers $a_1, a_2, a_3, \\ldots$ is chosen so that $a_n = a_{n - 1} - a_{n - 2}$ for each $n \\ge 3$. What is the sum of the first 2001 terms of this sequence if the sum of the first 1492 terms is 1985, and the sum of the first 1985 terms is 1492? Let's think ste... | math | {
"style": "rule",
"ground_truth": "986"
} | {
"split": "train",
"index": 28455,
"solution": "The problem gives us a sequence defined by a recursion, so let's calculate a few values to get a feel for how it acts. We aren't given initial values, so let $a_1 = a$ and $a_2 = b$. Then $a_3 = b - a$, $a_4 = (b - a) - b = -a$, $a_5 = -a - (b - a) = -b$, $a_6 = -b... |
deepscaler | [
{
"role": "user",
"content": "Let $p$, $q$, and $r$ be solutions of the equation $x^3 - 6x^2 + 11x = 14$.\n\nCompute $\\frac{pq}{r} + \\frac{qr}{p} + \\frac{rp}{q}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{47}{14}"
} | {
"split": "train",
"index": 28456,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $C: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$ with its right focus at $(\\sqrt{3}, 0)$, and passing through the point $(-1, \\frac{\\sqrt{3}}{2})$. Point $M$ is on the $x$-axis, and the line $l$ passing through $M$ intersects the ellipse $C$ at poin... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{21}}{21}"
} | {
"split": "train",
"index": 28457,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p,$ $q,$ $r,$ $s$ be real numbers such that\n\\[\\frac{(p - q)(r - s)}{(q - r)(s - p)} = \\frac{3}{7}.\\]Find the sum of all possible values of\n\\[\\frac{(p - r)(q - s)}{(p - q)(r - s)}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{4}"
} | {
"split": "train",
"index": 28458,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given Angie and Carlos are seated at a round table with three other people, determine the probability that Angie and Carlos are seated directly across from each other. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 28459,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area enclosed by the graph of $|x| + |3y| = 9$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 28460,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(\\alpha\\) and \\(\\beta\\) be angles such that \n\\[\n\\frac{\\cos^2 \\alpha}{\\cos \\beta} + \\frac{\\sin^2 \\alpha}{\\sin \\beta} = 2,\n\\]\nFind the sum of all possible values of \n\\[\n\\frac{\\sin^2 \\beta}{\\sin \\alpha} + \\frac{\\cos^2 \\beta}{\\cos \\alpha}.\n\... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 28461,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate\n\n(1) Use a simplified method to calculate $2017^{2}-2016 \\times 2018$;\n\n(2) Given $a+b=7$ and $ab=-1$, find the values of $(a+b)^{2}$ and $a^{2}-3ab+b^{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 28462,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A boulevard has 25 houses on each side, for a total of 50 houses. The addresses on the east side of the boulevard follow an arithmetic sequence, as do the addresses on the west side. On the east side, the addresses start at 5 and increase by 7 (i.e., 5, 12, 19, etc.), while on ... | math | {
"style": "rule",
"ground_truth": "113"
} | {
"split": "train",
"index": 28463,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $\\tan A = \\frac{1}{4}$ and $\\tan B = \\frac{3}{5}$.\n(1) Find the measure of angle $C$;\n(2) If the shortest side length of $\\triangle ABC$ is $\\sqrt{2}$, find the area of $\\triangle ABC$. Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 28464,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What percent of square $EFGH$ is shaded? All angles in the diagram are right angles. [asy]\nimport graph;\ndefaultpen(linewidth(0.7));\n\nxaxis(0,6,Ticks(1.0,NoZero));\nyaxis(0,6,Ticks(1.0,NoZero));\n\nfill((0,0)--(2,0)--(2,2)--(0,2)--cycle);\nfill((3,0)--(4,0)--(4,4)--(0,4)--(... | math | {
"style": "rule",
"ground_truth": "61.11\\%"
} | {
"split": "train",
"index": 28465,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line with slope $k$ intersects the parabola $y^{2}=4x$ at points $A$ and $B$, and is tangent to the circle $(x-5)^{2}+y^{2}=9$ at point $M$, where $M$ is the midpoint of segment $AB$. Find the value of $k$. 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": 28466,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $d$ be a positive number such that when $145$ is divided by $d$, the remainder is $4.$ Compute the sum of all possible two-digit values of $d$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 28467,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The quadratic equation $ax^2+8x+c=0$ has exactly one solution. If $a+c=10$, and $a<c$ find the ordered pair $(a,c)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(2,8)"
} | {
"split": "train",
"index": 28468,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "I have a drawer with 6 forks, 6 spoons, and 6 knives in it. If I reach in and randomly remove three pieces of silverware, what is the probability that I get one fork, one spoon, and one knife? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{34}"
} | {
"split": "train",
"index": 28469,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of \\(\\frac{\\log_2 80}{\\log_{40}2}-\\frac{\\log_2 160}{\\log_{20}2}\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 28470,
"solution": "We start with the given expression:\n\\[\n\\frac{\\log_2 80}{\\log_{40}2}-\\frac{\\log_2 160}{\\log_{20}2}\n\\]\n\nFirst, we use the change of base formula for logarithms, which states that $\\log_b a = \\frac{1}{\\log_a b}$. Applying this, we have:\n\\[\n\\log_{40... |
deepscaler | [
{
"role": "user",
"content": "Let $s$ be a table composed of positive integers (the table may contain the same number). Among the numbers in $s$ there is the number 68. The arithmetic mean of all numbers in $s$ is 56. However, if 68 is removed, the arithmetic mean of the remaining numbers drops to 55. What ... | math | {
"style": "rule",
"ground_truth": "649"
} | {
"split": "train",
"index": 28471,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given points $M(4,0)$ and $N(1,0)$, any point $P$ on curve $C$ satisfies: $\\overset{→}{MN} \\cdot \\overset{→}{MP} = 6|\\overset{→}{PN}|$.\n\n(I) Find the trajectory equation of point $P$;\n\n(II) A line passing through point $N(1,0)$ intersects curve $C$ at points $A$ and $B$... | math | {
"style": "rule",
"ground_truth": "-\\frac{8}{3}"
} | {
"split": "train",
"index": 28472,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a diagram, there is an equilateral triangle with a side length of $10$ m. Calculate both the perimeter and the height of the triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5\\sqrt{3}"
} | {
"split": "train",
"index": 28473,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $z$ is a complex number such that $z+\\frac 1z=2\\cos 3^\\circ$, find the least integer that is greater than $z^{2000}+\\frac 1{z^{2000}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 28474,
"solution": "Let $z=re^{i\\theta}$. Notice that we have $2\\cos(3^{\\circ})=e^{i\\frac{\\pi}{60}}+e^{-i\\frac{\\pi}{60}}=re^{i\\theta}+\\frac{1}{r}e^{-i\\theta}.$\n$r$ must be $1$ (or else if you take the magnitude would not be the same). Therefore, $z=e^{i\\frac{\\pi}{\\theta}... |
deepscaler | [
{
"role": "user",
"content": "Find the greatest value of $t$ such that \\[\\frac{t^2 - t -56}{t-8} = \\frac{3}{t+5}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 28475,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What three-digit number with units digit 2 and hundreds digit 4 is divisible by 9? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "432"
} | {
"split": "train",
"index": 28476,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let S$_{n}$ denote the sum of the first $n$ terms of the arithmetic sequence {a$_{n}$} with a common difference d=2. The terms a$_{1}$, a$_{3}$, and a$_{4}$ form a geometric sequence. Find the value of S$_{8}$. Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"split": "train",
"index": 28477,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that a triangular corner with side lengths DB=EB=1.5 is cut from an equilateral triangle ABC of side length 4.5, determine the perimeter of the remaining quadrilateral. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 28478,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the dihedral angle $\\alpha-l-\\beta$ is $60^{\\circ}$, points $P$ and $Q$ are on planes $\\alpha$ and $\\beta$ respectively. The distance from $P$ to plane $\\beta$ is $\\sqrt{3}$, and the distance from $Q$ to plane $\\alpha$ is $2 \\sqrt{3}$. What is the minimum di... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 28479,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $x \\in (0, \\frac{1}{2})$, find the minimum value of $\\frac{2}{x} + \\frac{9}{1-2x}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 28480,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse C: $$\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$$ (where $a>b>0$) that passes through the point $E(\\sqrt 3, 1)$, with an eccentricity of $\\frac{\\sqrt{6}}{3}$, and O as the origin of the coordinate system.\n(Ⅰ) Find the equation of the ellipse C.\n(Ⅱ) If point P... | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}"
} | {
"split": "train",
"index": 28481,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A firecracker was thrown vertically upwards with a speed of $20 \\mathrm{~m/s}$. Three seconds after the start of the flight, it exploded into two fragments of equal mass. The first fragment flew horizontally immediately after the explosion with a speed of $48 \\mathrm{~m/s}$. ... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 28482,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $a > 0$, $b > 0$, and it satisfies the equation $3a + b = a^2 + ab$. Find the minimum value of $2a + b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2} + 3"
} | {
"split": "train",
"index": 28483,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram below, $\\|\\overrightarrow{OA}\\| = 1,$ $\\|\\overrightarrow{OB}\\| = 1,$ and $\\|\\overrightarrow{OC}\\| = \\sqrt{2}.$ Also, $\\tan \\angle AOC = 7$ and $\\angle BOC = 45^\\circ.$\n\n[asy]\nunitsize(2 cm);\n\npair A, B, C, O;\n\nA = (1,0);\nB = (-0.6,0.8);\nC ... | math | {
"style": "rule",
"ground_truth": "\\left( \\frac{5}{4}, \\frac{7}{4} \\right)"
} | {
"split": "train",
"index": 28484,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A weather forecast predicts that for each day from Monday to Friday this week, there is a $40\\%$ chance of sun, a $25\\%$ chance of getting 4 inches of rain, and a $35\\%$ chance of getting 10 inches of rain. What is the expected value of the total number of inches of rain whi... | math | {
"style": "rule",
"ground_truth": "22.5"
} | {
"split": "train",
"index": 28485,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate: $(\\sqrt{3}+\\sqrt{2})(\\sqrt{3}-\\sqrt{2})^{2}=\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}-\\sqrt{2}"
} | {
"split": "train",
"index": 28486,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Our club has 10 members, and wishes to pick a president, secretary, treasurer, and morale officer. In how many ways can we choose the officers, if individual members can only hold at most one office? Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "5,\\!040"
} | {
"split": "train",
"index": 28487,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular table of dimensions \\( x \\) cm \\(\\times 80\\) cm is covered with identical sheets of paper of size \\( 5 \\) cm \\(\\times 8 \\) cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed one centimeter higher and one centimet... | math | {
"style": "rule",
"ground_truth": "77"
} | {
"split": "train",
"index": 28488,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular hexahedron with an edge length of $1$ is cut by planes passing through the common vertex of three edges and their respective midpoints. After removing the $8$ triangular pyramids, the volume of the remaining convex polyhedron is $\\_\\_\\_\\_\\_\\_$. Let's think step ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"split": "train",
"index": 28489,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let an integer be such that when divided by 20, the remainder is 11. What is the sum of the remainders when the same integer is divided by 4 and by 5? Additionally, find the smallest such integer greater than 50. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 28490,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three dice are thrown, and the sums of the points that appear on them are counted. In how many ways can you get a total of 5 points and 6 points? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 28491,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x$ and $y$ are positive integers less than 20 for which $x + y + xy = 99$, what is the value of $x + y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 28492,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the set \\( M = \\{1, 2, \\cdots, 2008\\} \\) of the first 2008 positive integers, a \\( k \\)-element subset \\( A \\) is chosen such that the sum of any two numbers in \\( A \\) cannot be divisible by the difference of those two numbers. What is the maximum value of \\( ... | math | {
"style": "rule",
"ground_truth": "670"
} | {
"split": "train",
"index": 28493,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ that satisfies: $a_{n+2}=\\begin{cases} 2a_{n}+1 & (n=2k-1, k\\in \\mathbb{N}^*) \\\\ (-1)^{\\frac{n}{2}} \\cdot n & (n=2k, k\\in \\mathbb{N}^*) \\end{cases}$, with $a_{1}=1$ and $a_{2}=2$, determine the maximum value of $n$ for which $S_n \\leqslan... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 28494,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $F$ is the right focus of the hyperbola $C$: $x^{2}- \\frac{y^{2}}{8}=1$, and $P$ is a point on the left branch of $C$, $A(0,6 \\sqrt {6})$ is a point on the $y$-axis. The minimum area of $\\triangle APF$ is $\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "6+9 \\sqrt {6}"
} | {
"split": "train",
"index": 28495,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the sum of $5.46$, $2.793$, and $3.1$ as a decimal. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11.353"
} | {
"split": "train",
"index": 28496,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A palindrome is an integer that reads the same forward and backward, such as 1221. What percent of the palindromes between 1000 and 2000 contain at least one digit 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10\\%"
} | {
"split": "train",
"index": 28497,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle has a side of length 6 cm, a side of length 8 cm and a right angle. What is the shortest possible length of the remaining side of the triangle? Express your answer in centimeters as a decimal to the nearest hundredth. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "5.29"
} | {
"split": "train",
"index": 28498,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCDE$ be a convex pentagon with $AB \\parallel CE, BC \\parallel AD, AC \\parallel DE, \\angle ABC=120^\\circ, AB=3, BC=5,$ and $DE = 15.$ Given that the ratio between the area of triangle $ABC$ and the area of triangle $EBD$ is $m/n,$ where $m$ and $n$ are relatively pri... | math | {
"style": "rule",
"ground_truth": "484"
} | {
"split": "train",
"index": 28499,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the difference between $1234_8$ and $765_8$ in base $8$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "225_8"
} | {
"split": "train",
"index": 28500,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At the start of this month, Mathilde and Salah each had 100 coins. For Mathilde, this was 25% more coins than she had at the start of last month. For Salah, this was 20% fewer coins than he had at the start of last month. What was the total number of coins that they had at the ... | math | {
"style": "rule",
"ground_truth": "205"
} | {
"split": "train",
"index": 28501,
"solution": "Suppose that Mathilde had $m$ coins at the start of last month and Salah had $s$ coins at the start of last month. From the given information, 100 is 25% more than $m$, so $100=1.25m$ which means that $m=\\frac{100}{1.25}=80$. From the given information, 100 is 20%... |
deepscaler | [
{
"role": "user",
"content": "A two-digit positive integer is said to be $cuddly$ if it is equal to the sum of its nonzero tens digit and the square of its units digit. How many two-digit positive integers are cuddly? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 28502,
"solution": "Let's analyze the problem step by step:\n\n1. **Define the Problem:**\n A two-digit integer is represented as $10a + b$, where $a$ is the tens digit and $b$ is the units digit. The integer is called $\\emph{cuddly}$ if it satisfies the equation:\n \\[\n 10a +... |
deepscaler | [
{
"role": "user",
"content": "In the expansion of $({\\frac{1}{x}-\\sqrt{x}})^{10}$, determine the coefficient of $x^{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 28503,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Tyler rolls two $ 4025 $ sided fair dice with sides numbered $ 1, \\dots , 4025 $ . Given that the number on the first die is greater than or equal to the number on the second die, what is the probability that the number on the first die is less than or equal to $ 2012 $ ? Le... | math | {
"style": "rule",
"ground_truth": "1006/4025"
} | {
"split": "train",
"index": 28504,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = x^2-3x$. For what values of $x$ is $f(f(x)) = f(x)$? Enter all the solutions, separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0, 3, -1, 4"
} | {
"split": "train",
"index": 28505,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cell phone plan costs $20$ dollars each month, plus $5$ cents per text message sent, plus $10$ cents for each minute used over $30$ hours. In January Michelle sent $100$ text messages and talked for $30.5$ hours. How much did she have to pay? Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "28.00"
} | {
"split": "train",
"index": 28506,
"solution": "1. **Calculate the base cost of the plan:**\n The base price of Michelle's cell phone plan is $20$ dollars.\n\n2. **Calculate the cost for text messages:**\n Michelle sent $100$ text messages, and each text message costs $5$ cents. Therefore, the total cost for... |
deepscaler | [
{
"role": "user",
"content": "Medians $BD$ and $CE$ of triangle $ABC$ are perpendicular, $BD=8$, and $CE=12$. The area of triangle $ABC$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 28507,
"solution": "Given that medians $BD$ and $CE$ of triangle $ABC$ are perpendicular, and their lengths are $BD = 8$ and $CE = 12$, we need to find the area of triangle $ABC$.\n\n1. **Understanding the properties of medians**: \n - The medians of a triangle intersect at the cent... |
deepscaler | [
{
"role": "user",
"content": "Given the line y=b intersects with the function f(x)=2x+3 and the function g(x)=ax+ln x (where a is a real constant in the interval [0, 3/2]), find the minimum value of |AB|. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 - \\frac{\\ln 2}{2}"
} | {
"split": "train",
"index": 28508,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $g(x) = 3x^4 + 2x^3 - x^2 - 4x + s$. For what value of $s$ is $g(-1) = 0$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 28509,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "As part of his effort to take over the world, Edward starts producing his own currency. As part of an effort to stop Edward, Alex works in the mint and produces 1 counterfeit coin for every 99 real ones. Alex isn't very good at this, so none of the counterfeit coins are the rig... | math | {
"style": "rule",
"ground_truth": "\\frac{19}{28}"
} | {
"split": "train",
"index": 28510,
"solution": "$5 \\%$ of the coins are sent to the lab, and only $.95 \\%$ of the coins are sent to the lab and counterfeit, so there is a $19 \\%$ chance that a coin sent to the lab is counterfeit and an $81 \\%$ chance that it is real. The lab could correctly detect a counterf... |
deepscaler | [
{
"role": "user",
"content": "On the segment $OA$ of length $L$ on the number line $Ox$, two points, $B(x)$ and $C(y)$, are randomly placed. Find the probability that from the three resulting segments a triangle can be formed. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/4"
} | {
"split": "train",
"index": 28511,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two tigers, Alice and Betty, run in the same direction around a circular track with a circumference of 400 meters. Alice runs at a speed of \\(10 \\, \\text{m/s}\\) and Betty runs at \\(15 \\, \\text{m/s}\\). Betty gives Alice a 40 meter head start before they both start runnin... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 28512,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the shortest distance from a point on the curve $y=x^{2}-\\ln x$ to the line $x-y-2=0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 28513,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate: \\(\\left[\\left(11 \\frac{1}{9}-3 \\frac{2}{5} \\times 1 \\frac{2}{17}\\right)-8 \\frac{2}{5} \\div 3.6\\right] \\div 2 \\frac{6}{25}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{20}{9}"
} | {
"split": "train",
"index": 28514,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a hyperbola $C$ with one of its foci on the line $l: 4x-3y+20=0$, and one of its asymptotes is parallel to $l$, and the foci of the hyperbola $C$ are on the $x$-axis, then the standard equation of the hyperbola $C$ is \\_\\_\\_\\_\\_\\_; the eccentricity is \\_\\_\\_\\_\\... | math | {
"style": "rule",
"ground_truth": "\\dfrac{5}{3}"
} | {
"split": "train",
"index": 28515,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Inside triangle \\(ABC\\), a point \\(O\\) is chosen such that \\(\\angle ABO = \\angle CAO\\), \\(\\angle BAO = \\angle BCO\\), and \\(\\angle BOC = 90^{\\circ}\\). Find the ratio \\(AC : OC\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 28516,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If each digit of a four-digit natural number $M$ is not $0$, and the five times of the digit in the thousandth place is equal to the sum of the digits in the hundredth, tenth, and unit places, then this four-digit number is called a \"modest number\". For example, for the four-... | math | {
"style": "rule",
"ground_truth": "3816"
} | {
"split": "train",
"index": 28517,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jing purchased eight identical items. If the total cost was $\\$ 26$, what is the cost per item, in dollars? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{26}{8}"
} | {
"split": "train",
"index": 28518,
"solution": "Since Jing purchased 8 identical items and the total cost was $\\$ 26$, then to obtain the cost per item, she divides the total cost by the number of items. Thus, the answer is $26 \\div 8$."
} |
deepscaler | [
{
"role": "user",
"content": "Joe has a collection of $23$ coins, consisting of $5$-cent coins, $10$-cent coins, and $25$-cent coins. He has $3$ more $10$-cent coins than $5$-cent coins, and the total value of his collection is $320$ cents. How many more $25$-cent coins does Joe have than $5$-cent coins? Le... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 28519,
"solution": "Let $n$, $d$, and $q$ represent the number of $5$-cent coins, $10$-cent coins, and $25$-cent coins in Joe's collection, respectively. We are given the following equations based on the problem statement:\n\\[\nn + d + q = 23 \\quad \\text{(1)}\n\\]\n\\[\n5n + 10d + ... |
deepscaler | [
{
"role": "user",
"content": "The radian measure of -150° is equal to what fraction of π. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{5\\pi}{6}"
} | {
"split": "train",
"index": 28520,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a $123 \\times 123$ board, each cell is painted either purple or blue according to the following conditions:\n\n- Each purple cell that is not on the edge of the board has exactly 5 blue cells among its 8 neighbors.\n- Each blue cell that is not on the edge of the board has ... | math | {
"style": "rule",
"ground_truth": "6724"
} | {
"split": "train",
"index": 28521,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For all integers $n$ greater than 1, define $a_n =\n\\dfrac{1}{\\log_n 2002}$. Let $b = a_2 + a_3 + a_4 + a_5$ and $c=\na_{10} + a_{11} + a_{12} + a_{13} + a_{14}$. Find $b - c.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 28522,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given four distinct real numbers \\( a, b, c, d \\) such that \\(\\frac{a}{b} + \\frac{b}{c} + \\frac{c}{d} + \\frac{d}{a} = 4\\) and \\( ac = bd \\), find the maximum value of \\(\\frac{a}{c} + \\frac{b}{d} + \\frac{c}{a} + \\frac{d}{b} \\). Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "-12"
} | {
"split": "train",
"index": 28523,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given 5 differently colored balls and 3 different boxes, with the requirement that no box is empty, calculate the total number of different ways to place 4 balls into the boxes. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "180"
} | {
"split": "train",
"index": 28524,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$, then the average value (arithmetic mean) of the integers remaining is $32$. If the least integer in $S$ is also removed, then the average value of the integers remaining i... | math | {
"style": "rule",
"ground_truth": "36.8"
} | {
"split": "train",
"index": 28525,
"solution": "Let $S = \\{a_1, a_2, a_3, \\hdots, a_n\\}$ where $a_1 < a_2 < a_3 < \\hdots < a_n$. We are given the following conditions:\n1. Removing the greatest integer $a_n$ from $S$ results in an average of $32$ for the remaining integers.\n2. Removing both the greatest int... |
deepscaler | [
{
"role": "user",
"content": "How many positive integers less than $900$ are either a perfect cube or a perfect square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 28526,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sum of the polynomials $A$ and $B$ is $12x^{2}y+2xy+5$, where $B=3x^{2}y-5xy+x+7$. Find:<br/>$(1)$ The polynomial $A$;<br/>$(2)$ When $x$ takes any value, the value of the expression $2A-\\left(A+3B\\right)$ is a constant. Find the value of $y$. Let's think step ... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{11}"
} | {
"split": "train",
"index": 28527,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of positive integers $n$ less than $1000$ for which there exists a positive real number $x$ such that $n=x\\lfloor x \\rfloor$.\n Note: $\\lfloor x \\rfloor$ is the greatest integer less than or equal to $x$. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "496"
} | {
"split": "train",
"index": 28528,
"solution": "Bounding gives $x^2\\le n<x^2+x$. Thus there are a total of $x$ possible values for $n$, for each value of $x^2$. Checking, we see $31^2+31=992<1000$, so there are \\[1+2+3+...+32= \\boxed{496}\\] such values for $n$."
} |
deepscaler | [
{
"role": "user",
"content": "The median of a set of consecutive odd integers is 138. If the greatest integer in the set is 145, what is the smallest integer in the set? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "131"
} | {
"split": "train",
"index": 28529,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The rules for a race require that all runners start at $A$, touch any part of the 1200-meter wall, and stop at $B$. What is the number of meters in the minimum distance a participant must run? Express your answer to the nearest meter. [asy]\nimport olympiad; import geometry; si... | math | {
"style": "rule",
"ground_truth": "1442"
} | {
"split": "train",
"index": 28530,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the maximum value of the expression $-5r^2 + 40r - 12$ for real $r$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "68"
} | {
"split": "train",
"index": 28531,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cone is inverted and filled with water to 3/4 of its height. What percent of the cone's volume is filled with water? Express your answer as a decimal to the nearest ten-thousandth. (You should enter 10.0000 for $10\\%$ instead of 0.1000.) Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "42.1875"
} | {
"split": "train",
"index": 28532,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence \\( b_1, b_2, b_3, \\ldots \\) is defined recursively by \\( b_1 = 2 \\), \\( b_2 = 3 \\), and for \\( k \\geq 3 \\),\n\\[ b_k = \\frac{1}{2} b_{k-1} + \\frac{1}{3} b_{k-2}. \\]\nEvaluate \\( b_1 + b_2 + b_3 + \\dotsb. \\) Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 28533,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a triangle $ABC$ with internal angles $A$, $B$, and $C$, and centroid $G$. If $2\\sin A\\overrightarrow{GA}+\\sqrt{3}\\sin B\\overrightarrow{GB}+3\\sin C\\cdot \\overrightarrow{GC}=\\vec{0}$, then $\\cos B=$_______. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "\\dfrac {1}{12}"
} | {
"split": "train",
"index": 28534,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The English alphabet, which has 26 letters, is randomly permuted. Let \\(p_{1}\\) be the probability that \\(\\mathrm{AB}, \\mathrm{CD}\\), and \\(\\mathrm{EF}\\) all appear as contiguous substrings. Let \\(p_{2}\\) be the probability that \\(\\mathrm{ABC}\\) and \\(\\mathrm{DE... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 28535,
"solution": "There are 23! ways to arrange the alphabet such that AB, CD, and EF all appear as contiguous substrings: treat each of these pairs of letters as a single merged symbol, which leaves 23 symbols to permute. Similarly, there are 22! ways to arrange the alphabet such t... |
deepscaler | [
{
"role": "user",
"content": "In an opaque bag, there are $10$ balls each of red, white, and yellow colors, all identical except for the color. At least how many balls must be drawn to ensure that two balls of different colors are drawn? At least how many balls must be drawn to ensure that two yellow balls ... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 28536,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, it is known that the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, $c$, with $a=\\sqrt{6}$, $\\sin ^{2}B+\\sin ^{2}C=\\sin ^{2}A+\\frac{2\\sqrt{3}}{3}\\sin A\\sin B\\sin C$. Choose one of the following conditions to determine whether tri... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 28537,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( f(n) \\) be the integer closest to \\( \\sqrt[4]{n} \\). Then, \\( \\sum_{k=1}^{2018} \\frac{1}{f(k)} = \\) ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2823}{7}"
} | {
"split": "train",
"index": 28538,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Square $ABCD$ has sides of length 4, and $M$ is the midpoint of $\\overline{CD}$. A circle with radius 2 and center $M$ intersects a circle with radius 4 and center $A$ at points $P$ and $D$. What is the distance from $P$ to $\\overline{AD}$? Express your answer as a common fra... | math | {
"style": "rule",
"ground_truth": "\\frac{16}{5}"
} | {
"split": "train",
"index": 28539,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When each of $702$, $787$, and $855$ is divided by the positive integer $m$, the remainder is always the positive integer $r$. When each of $412$, $722$, and $815$ is divided by the positive integer $n$, the remainder is always the positive integer $s \\neq r$. Find $m+n+r+s$. ... | math | {
"style": "rule",
"ground_truth": "62"
} | {
"split": "train",
"index": 28540,
"solution": "We know that $702 = am + r, 787 = bm + r,$ and $855 = cm+r$ where $a-c$ are integers.\n Subtracting the first two, the first and third, and the last two, we get $85 = (b-a)m, 153=(c-a)m,$ and $68=(c-b)m.$\n We know that $b-a, c-a$ and $c-b$ must be integers, so all... |
deepscaler | [
{
"role": "user",
"content": "A rectangle has a perimeter of 80 cm and each side has an integer length. How many non-congruent rectangles meet these criteria? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 28541,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The inclination angle of the line $\\sqrt{3}x + y - 1 = 0$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"split": "train",
"index": 28542,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $a-b=4$ and $b+c=2$, determine the value of $a^2+b^2+c^2-ab+bc+ca$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 28543,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mary divides a circle into 12 sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle? Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 28544,
"solution": "1. **Define the problem in terms of an arithmetic sequence:** Let the central angles of the sectors be represented by an arithmetic sequence with the first term $a_1$ and common difference $d$. The sequence will have 12 terms, and the sum of these terms must equal ... |
deepscaler | [
{
"role": "user",
"content": "Given the line $y=x+\\sqrt{6}$, the circle $(O)$: $x^2+y^2=5$, and the ellipse $(E)$: $\\frac{y^2}{a^2}+\\frac{x^2}{b^2}=1$ $(b > 0)$ with an eccentricity of $e=\\frac{\\sqrt{3}}{3}$. The length of the chord intercepted by line $(l)$ on circle $(O)$ is equal to the length of th... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 28545,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$P, A, B, C,$ and $D$ are five distinct points in space such that $\\angle APB = \\angle BPC = \\angle CPD = \\angle DPA = \\theta$, where $\\theta$ is a given acute angle. Determine the greatest and least values of $\\angle APC + \\angle BPD$. Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "0^\\circ \\text{ and } 360^\\circ"
} | {
"split": "train",
"index": 28546,
"solution": "\nConsider five distinct points \\( P, A, B, C, \\) and \\( D \\) in space where the angles formed at \\( P \\) satisfy \\( \\angle APB = \\angle BPC = \\angle CPD = \\angle DPA = \\theta \\). We are tasked with finding the greatest and least possible values of the... |
deepscaler | [
{
"role": "user",
"content": "A four-meter gas pipe has rusted in two places. Determine the probability that all three resulting parts can be used as connections to gas stoves, if according to standards, a stove should not be located closer than 1 meter from the main gas pipe. Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "1/4"
} | {
"split": "train",
"index": 28547,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, points $D$ and $E$ lie on $\\overline{BC}$ and $\\overline{AC}$ respectively. Lines $\\overline{AD}$ and $\\overline{BE}$ intersect at point $T$ such that $AT/DT=2$ and $BT/ET=3$. Determine the ratio $CD/BD$. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"split": "train",
"index": 28548,
"solution": ""
} |
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