data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "There are 2006 positive integers \\( a_{1}, a_{2}, \\cdots, a_{2006} \\) (which can be the same) such that the ratios \\( \\frac{a_{1}}{a_{2}}, \\frac{a_{2}}{a_{3}}, \\cdots, \\frac{a_{2005}}{a_{2006}} \\) are all distinct. What is the minimum number of distinct numbers among \... | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 8800,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many 9-digit numbers that are divisible by 5 can be formed by permuting the digits of the number 377353752? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1120"
} | {
"split": "train",
"index": 8801,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\sin \\theta= \\frac {3}{5}$ and $\\frac {5\\pi}{2} < \\theta < 3\\pi$, then $\\sin \\frac {\\theta}{2}=$ ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac {3 \\sqrt {10}}{10}"
} | {
"split": "train",
"index": 8802,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The positive integers $A, B$ and $C$ form an arithmetic sequence while the integers $B, C$ and $D$ form a geometric sequence. If $\\frac CB = \\frac 53,$ what is the smallest possible value of $A + B + C + D$? Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 8803,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute \\[\\lfloor 1 \\rfloor + \\lfloor 1.6 \\rfloor + \\lfloor 2.2 \\rfloor + \\lfloor 2.8 \\rfloor + \\dots + \\lfloor 99.4 \\rfloor + \\lfloor 100 \\rfloor,\\]where the arguments of the floor functions are in arithmetic progression. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "8317"
} | {
"split": "train",
"index": 8804,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of\n\\[x^2 + 2xy + 3y^2 - 6x - 2y,\\]over all real numbers $x$ and $y.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-11"
} | {
"split": "train",
"index": 8805,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the measure of the smaller angle between the hands of a 12-hour clock at 12:25 pm, in degrees? Express your answer as a decimal to the nearest tenth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "137.5\\text{ degrees}"
} | {
"split": "train",
"index": 8806,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the distances from one vertex of a rectangle with length $3$ and width $4$ to the centers of the opposite sides. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{13} + 2"
} | {
"split": "train",
"index": 8807,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $a_2+a_4+a_6+a_8+\\ldots+a_{98}$ if $a_1$, $a_2$, $a_3\\ldots$ is an arithmetic progression with common difference 1, and $a_1+a_2+a_3+\\ldots+a_{98}=137$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "93"
} | {
"split": "train",
"index": 8808,
"solution": "Since we are dealing with an arithmetic sequence, \\[a_2+a_4+a_6+a_8+\\ldots+a_{98} = 49a_{50}\\] We can also figure out that \\[a_1+a_2+a_3+\\ldots+a_{98} = a_1 + 97a_{50} = 137\\] \\[a_1 = a_{50}-49 \\Rightarrow 98a_{50}-49 = 137\\] Thus, $49a_{50} = \\frac{137 + ... |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive real number $x$ such that\n\\[\\lfloor x^2 \\rfloor - x \\lfloor x \\rfloor = 10.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{131}{11}"
} | {
"split": "train",
"index": 8809,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the smallest positive angle $x,$ in degrees, such that\n\\[\\tan 4x = \\frac{\\cos x - \\sin x}{\\cos x + \\sin x}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9^\\circ"
} | {
"split": "train",
"index": 8810,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An amusement park has a collection of scale models, with a ratio of $1: 20$, of buildings and other sights from around the country. The height of the United States Capitol is $289$ feet. What is the height in feet of its duplicate to the nearest whole number? Let's think step b... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 8811,
"solution": "1. **Identify the scale ratio and the actual height of the building**: The scale model is built at a ratio of $1:20$, and the actual height of the United States Capitol is $289$ feet.\n\n2. **Calculate the height of the scale model**: To find the height of the scale... |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathrm{C}$ be a circle in the $\\mathrm{xy}$-plane with a radius of 1 and its center at $O(0,0,0)$. Consider a point $\\mathrm{P}(3,4,8)$ in space. If a sphere is completely contained within the cone with $\\mathrm{C}$ as its base and $\\mathrm{P}$ as its apex, find the ... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}\\pi(3-\\sqrt{5})^3"
} | {
"split": "train",
"index": 8812,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)=x^2-7x+18$ and let $g(f(x))=2x+3$. What is the sum of all possible values of $g(8)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 8813,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{A} = \\begin{pmatrix} 1 & 2 \\\\ 3 & 4 \\end{pmatrix}$ and $\\mathbf{B} = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}$ be two matrices such that $\\mathbf{A} \\mathbf{B} = \\mathbf{B} \\mathbf{A}.$ Assuming $3b \\neq c,$ find $\\frac{a - d}{c - 3b}.$ Let's t... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 8814,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If set $A=\\{-4, 2a-1, a^2\\}$, $B=\\{a-5, 1-a, 9\\}$, and $A \\cap B = \\{9\\}$, then the value of $a$ is. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 8815,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the equation \"Xiwangbei jiushi hao $\\times$ 8 = Jiushihao Xiwangbei $\\times$ 5\", different Chinese characters represent different digits. The six-digit even number represented by \"Xiwangbei jiushi hao\" is ____. Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "256410"
} | {
"split": "train",
"index": 8816,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$\\alpha$ and $\\beta$ are two parallel planes. Four points are taken within plane $\\alpha$, and five points are taken within plane $\\beta$.\n(1) What is the maximum number of lines and planes that can be determined by these points?\n(2) What is the maximum number of tetrahed... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 8817,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers, including $1,$ are divisors of both $40$ and $72?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\mbox{four}"
} | {
"split": "train",
"index": 8818,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a Go championship participated by three players: A, B, and C, the matches are conducted according to the following rules: the first match is between A and B; the second match is between the winner of the first match and C; the third match is between the winner of the second ... | math | {
"style": "rule",
"ground_truth": "0.162"
} | {
"split": "train",
"index": 8819,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the integer $n$, $-90 \\le n \\le 90$, such that $\\sin n^\\circ = \\sin 782^\\circ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-62"
} | {
"split": "train",
"index": 8820,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the 308th digit to the right of the decimal point when $\\frac{12}{37}$ is expressed as a decimal? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 8821,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We say that an ordered pair $(a,b)$ of positive integers with $a>b$ is square-ish if both $a+b$ and $a-b$ are perfect squares. For example, $(17,8)$ is square-ish because $17+8=25$ and $17-8=9$ are both perfect squares. How many square-ish pairs $(a,b)$ with $a... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 8822,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ and $y$ be real numbers such that $\\frac{\\sin x}{\\sin y} = 3$ and $\\frac{\\cos x}{\\cos y} = \\frac12$. The value of $\\frac{\\sin 2x}{\\sin 2y} + \\frac{\\cos 2x}{\\cos 2y}$ can be expressed in the form $\\frac pq$, where $p$ and $q$ are relatively prime positive i... | math | {
"style": "rule",
"ground_truth": "107"
} | {
"split": "train",
"index": 8823,
"solution": "Let $a = \\sin(x), b = \\sin(y)$ The first equation yields $\\frac{a}{b} = 3.$ Using $sin^2(x) + cos^2(x) = 1$ the second equation yields \\[\\frac{\\sqrt{1-a^2}}{\\sqrt{1-b^2}} = \\frac{1}{2} \\rightarrow \\frac{1-a^2}{1-b^2} = \\frac{1}{4}\\]\n Solving this yields... |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $\\frac{b}{a}+\\sin({A-B})=\\sin C$. Find:<br/>\n$(1)$ the value of angle $A$;<br/>\n$(2)$ if $a=2$, find the maximum value of $\\sqrt{... | math | {
"style": "rule",
"ground_truth": "\\frac{12}{5}"
} | {
"split": "train",
"index": 8824,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the total surface area of a cone with a diameter of 8 cm and a height of 12 cm. Express your answer in terms of \\(\\pi\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16\\pi (\\sqrt{10} + 1)"
} | {
"split": "train",
"index": 8825,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\\\_a{n}\\_$, where $\\_a{n}>0$, $\\_a{1}=1$, and $\\_a{n+2}=\\frac{1}{a{n}+1}$, and it is known that $\\_a{6}=a{2}$, find the value of $\\_a{2016}+a{3}=\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{2}"
} | {
"split": "train",
"index": 8826,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "S is a subset of {1, 2, 3, ... , 16} which does not contain three integers which are relatively prime in pairs. How many elements can S have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 8827,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cedric has deposited $\\$12,\\!000$ into an account that pays $5\\%$ interest compounded annually.\n\nDaniel has deposited $\\$12,\\!000$ into an account that pays $7\\%$ simple annual interest.\n\nIn $15$ years Cedric and Daniel compare their respective balances. To the neares... | math | {
"style": "rule",
"ground_truth": "\\$347"
} | {
"split": "train",
"index": 8828,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Euler family has four girls aged $6$, $6$, $9$, and $11$, and two boys aged $13$ and $16$. What is the mean (average) age of the children? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{61}{6}"
} | {
"split": "train",
"index": 8829,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the six digits 0, 1, 2, 3, 4, 5, select two odd numbers and two even numbers to form a four-digit number without repeating digits. The total number of such four-digit numbers is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "180"
} | {
"split": "train",
"index": 8830,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alicia had two containers. The first was $\\frac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\\frac{3}{4}$ full of water. What is the ratio of the volume of ... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{10}"
} | {
"split": "train",
"index": 8831,
"solution": "\n#### Step 1: Set up the equation\nLet the volume of the first container be $A$ and the volume of the second container be $B$. According to the problem, $\\frac{5}{6}$ of the first container's volume is equal to $\\frac{3}{4}$ of the second container's volume when ... |
deepscaler | [
{
"role": "user",
"content": "Let $a$, $b$, $c$, $d$, and $e$ be positive integers with $a+b+c+d+e=2510$. Let $N$ be the largest of the sums $a+b$, $b+c$, $c+d$, and $d+e$. What is the smallest possible value of $N$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1255"
} | {
"split": "train",
"index": 8832,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a real number $x$ , let $f(x)=\\int_0^{\\frac{\\pi}{2}} |\\cos t-x\\sin 2t|\\ dt$ .\n\n(1) Find the minimum value of $f(x)$ .\n\n(2) Evaluate $\\int_0^1 f(x)\\ dx$ .\n\n*2011 Tokyo Institute of Technology entrance exam, Problem 2* Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4} + \\frac{1}{2} \\ln 2"
} | {
"split": "train",
"index": 8833,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equation \\( \\sin^2 x + \\sin^2 3x + \\sin^2 5x + \\sin^2 6x = \\frac{81}{32} \\) can be reduced to the equivalent equation\n\\[ \\cos ax \\cos bx \\cos cx = 0, \\] for some positive integers \\( a, \\) \\( b, \\) and \\( c. \\) Find \\( a + b + c. \\) Let's think step by ... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 8834,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the square root of $\\dfrac{10!}{210}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24\\sqrt{30}"
} | {
"split": "train",
"index": 8835,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $f(x)$ is an odd function on $\\mathbb{R}$ and satisfies $f(x+4)=f(x)$, when $x \\in (0,2)$, $f(x)=2x^2$. Evaluate $f(2015)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 8836,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We place planes through each edge and the midpoint of the edge opposite to it in a tetrahedron. Into how many parts do these planes divide the tetrahedron, and what are the volumes of these parts? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 8837,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\sqrt{8} \\times \\sqrt{50}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 8838,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are values $A$ and $B$ such that\n\\[\\frac{Bx-11}{x^2-7x+10}=\\frac{A}{x-2}+\\frac{3}{x-5}.\\]Find $A+B$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 8839,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given positive integers \\( x, y, z \\) that satisfy the condition \\( x y z = (14 - x)(14 - y)(14 - z) \\), and \\( x + y + z < 28 \\), what is the maximum value of \\( x^2 + y^2 + z^2 \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "219"
} | {
"split": "train",
"index": 8840,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If 260 were expressed as a sum of at least three distinct powers of 2, what would be the least possible sum of the exponents of these powers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 8841,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the number of quadruples $(a, b, c, d)$ of positive integers satisfying $12a+21b+28c+84d=2024$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2024"
} | {
"split": "train",
"index": 8842,
"solution": "Looking at the equation $\\bmod 7$ gives $a \\equiv 3(\\bmod 7)$, so let $a=7a^{\\prime}+3$. Then mod 4 gives $b \\equiv 0(\\bmod 4)$, so let $b=4b^{\\prime}$. Finally, $\\bmod 3$ gives $c \\equiv 2(\\bmod 3)$, so let $c=3c^{\\prime}+2$. Now our equation yields $$84... |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = x^4 + ax^3 + bx^2 + cx + d$ be a polynomial whose roots are all negative integers. If $a + b + c + d = 2009,$ find $d.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "528"
} | {
"split": "train",
"index": 8843,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers between $\\frac{23}{3}$ and $\\frac{65}{2}$ are multiples of $5$ or $3$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 8844,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the ordered pair $(a,b)$ of real numbers such that the cubic polynomials $x^3 + ax^2 + 11x + 6 = 0$ and $x^3 + bx^2 + 14x + 8 = 0$ have two distinct roots in common. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(6,7)"
} | {
"split": "train",
"index": 8845,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $\\cos \\frac{\\pi}{9} \\cdot \\cos \\frac{2\\pi}{9} \\cdot \\cos \\frac{4\\pi}{9} = $ ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 8846,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the length of the segment tangent from the origin to the circle that passes through the points $(4,5)$, $(8,10)$, and $(10,25)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{82}"
} | {
"split": "train",
"index": 8847,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x, y$, and $y-\\frac{1}{x}$ are not $0$, then $\\frac{x-\\frac{1}{y}}{y-\\frac{1}{x}}$ equals Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{x}{y}"
} | {
"split": "train",
"index": 8848,
"solution": "1. **Start with the given expression:**\n \\[\n \\frac{x-\\frac{1}{y}}{y-\\frac{1}{x}}\n \\]\n\n2. **Multiply the numerator and the denominator by $xy$ to eliminate the fractions:**\n \\[\n \\frac{x-\\frac{1}{y}}{y-\\frac{1}{x}} \\cdot \\frac{xy}{xy} = \\f... |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $a_n$: $\\frac{1}{1}$, $\\frac{2}{1}$, $\\frac{1}{2}$, $\\frac{3}{1}$, $\\frac{2}{2}$, $\\frac{1}{3}$, $\\frac{4}{1}$, $\\frac{3}{2}$, $\\frac{2}{3}$, $\\frac{1}{4}$, ..., according to the pattern of its first 10 terms, the value of $a_{99}+a_{100}$ is \\_\\_... | math | {
"style": "rule",
"ground_truth": "\\frac{37}{24}"
} | {
"split": "train",
"index": 8849,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the integers \\(19, 21, 23, 25, 27\\) can be expressed as the sum of two prime numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 8850,
"solution": "We note that all of the given possible sums are odd, and also that every prime number is odd with the exception of 2 (which is even). When two odd integers are added, their sum is even. When two even integers are added, their sum is even. When one even integer and o... |
deepscaler | [
{
"role": "user",
"content": "Solve \\[\\frac{x^2+x+1}{x+1}=x+2\\]for $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac12"
} | {
"split": "train",
"index": 8851,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Joshua rolls two dice and records the product of the numbers face up. The probability that this product is composite can be expressed as $\\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ . Compute $m+n$ .\n\n*Proposed by Nathan Xiong* Let's think step by s... | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 8852,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A polynomial with integer coefficients is of the form\n\\[x^3 + a_2 x^2 + a_1 x - 11 = 0.\\]Enter all the possible integer roots of this polynomial, separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-11, -1, 1, 11"
} | {
"split": "train",
"index": 8853,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $P$, $Q$, $R$, and $S$ are in space such that each of $\\overline{SP}$, $\\overline{SQ}$, and $\\overline{SR}$ is perpendicular to the other two. If $SP = SQ = 12$ and $SR = 7$, then what is the volume of pyramid $SPQR$? Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "168"
} | {
"split": "train",
"index": 8854,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system on the plane, establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of the $x$-axis as the polar axis. The polar coordinates of point $A$ are $\\left( 4\\sqrt{2}, \\frac{\\pi}{4} \\right)$, and... | math | {
"style": "rule",
"ground_truth": "\\frac{32}{7}"
} | {
"split": "train",
"index": 8855,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $PQR$ has side lengths $PQ=6$, $QR=8$, and $PR=9$. Two bugs start simultaneously from $P$ and crawl along the perimeter of the triangle in opposite directions at the same speed. They meet at point $S$. What is $QS$? Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "5.5"
} | {
"split": "train",
"index": 8856,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A solid box is 20 cm by 15 cm by 12 cm. A new solid is formed by removing a cube 4 cm on a side from each of the top four corners of this box. After that, four cubes, 2 cm on a side, are placed on each lower corner of the box. What percent of the original volume has been altere... | math | {
"style": "rule",
"ground_truth": "6.22\\%"
} | {
"split": "train",
"index": 8857,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $\\tan A= \\frac {3}{4}$ and $\\tan (A-B)=- \\frac {1}{3}$, find the value of $\\tan C$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {79}{3}"
} | {
"split": "train",
"index": 8858,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The largest three-digit number divided by an integer, with the quotient rounded to one decimal place being 2.5, will have the smallest divisor as: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "392"
} | {
"split": "train",
"index": 8859,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the relationship between shelf life and storage temperature is an exponential function $y = ka^x$, where milk has a shelf life of about $100$ hours in a refrigerator at $0°C$, and about $80$ hours in a refrigerator at $5°C$, determine the approximate shelf life of milk in... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 8860,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $(x,y,z)$ be an ordered triplet of real numbers that satisfies the following system of equations: \\begin{align*}x+y^2+z^4&=0,y+z^2+x^4&=0,z+x^2+y^4&=0.\\end{align*} If $m$ is the minimum possible value of $\\lfloor x^3+y^3+z^3\\rfloor$ , find the modulo $2007$ resid... | math | {
"style": "rule",
"ground_truth": "2004"
} | {
"split": "train",
"index": 8861,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of the region bounded by a function $y=-x^4+16x^3-78x^2+50x-2$ and the tangent line which is tangent to the curve at exactly two distinct points.\n\nProposed by Kunihiko Chikaya Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1296/5"
} | {
"split": "train",
"index": 8862,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When three standard dice are tossed, the numbers $a,b,c$ are obtained. Find the probability that $$(a-1)(b-1)(c-1) \\neq 0$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{125}{216}"
} | {
"split": "train",
"index": 8863,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A man has $2.73 in pennies, nickels, dimes, quarters and half dollars. If he has an equal number of coins of each kind, then the total number of coins he has is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 8864,
"solution": "Let $x$ represent the number of each type of coin the man has. Since he has pennies, nickels, dimes, quarters, and half dollars, and he has an equal number of each, we can set up the following equation to represent the total value of his coins in cents:\n\n- Pennies... |
deepscaler | [
{
"role": "user",
"content": "Suppose there are $160$ pigeons and $n$ holes. The $1$ st pigeon flies to the $1$ st hole, the $2$ nd pigeon flies to the $4$ th hole, and so on, such that the $i$ th pigeon flies to the $(i^2\\text{ mod }n)$ th hole, where $k\\text{ mod }n$ is the remainder when $... | math | {
"style": "rule",
"ground_truth": "326"
} | {
"split": "train",
"index": 8865,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, \\( S \\) lies on \\( R T \\), \\( \\angle Q T S = 40^{\\circ} \\), \\( Q S = Q T \\), and \\( \\triangle P R S \\) is equilateral. The value of \\( x \\) is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 8866,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence ${a_n}$ satisfies $a_1=1$, $a_{n+1} \\sqrt { \\frac{1}{a_{n}^{2}}+4}=1$. Let $S_{n}=a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}$. If $S_{2n+1}-S_{n}\\leqslant \\frac{m}{30}$ holds for any $n\\in\\mathbb{N}^{*}$, find the minimum value of the positive integer $m$. Let's think... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 8867,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Chinese Basketball Association (CBA) Finals, a best-of-seven series is played, where the first team to win four games is declared the champion. Team A and Team B are evenly matched in the finals, with each team having a probability of $\\frac{1}{2}$ of winning each game.... | math | {
"style": "rule",
"ground_truth": "3775"
} | {
"split": "train",
"index": 8868,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Amy works for 36 hours per week for 10 weeks during the summer, making $\\$3000$. If she works for 30 weeks during the school year at the same rate of pay and needs to make another $\\$3000$, how many hours per week must she work? Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 8869,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find a nonzero $p$ such that $px^2-12x+4=0$ has only one solution. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 8870,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f\\left( x \\right)=\\sin\\left( 2x+\\varphi \\right)+\\sqrt{3}\\cos\\left( 2x+\\varphi \\right)$ $\\left( 0 < \\varphi < \\pi \\right)$, its graph is shifted left by $\\frac{\\pi }{4}$ units, and the shifted graph is symmetric about the point $\\left( \\fra... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 8871,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On floor 0 of a weird-looking building, you enter an elevator that only has one button. You press the button twice and end up on floor 1. Thereafter, every time you press the button, you go up by one floor with probability $\\frac{X}{Y}$, where $X$ is your current floor, and $Y... | math | {
"style": "rule",
"ground_truth": "97"
} | {
"split": "train",
"index": 8872,
"solution": "By induction, we can determine that after $n$ total button presses, your current floor is uniformly distributed from 1 to $n-1$ : the base case $n=2$ is trivial to check, and for the $n+1$ th press, the probability that you are now on floor $i$ is $\\frac{1}{n-1}\\l... |
deepscaler | [
{
"role": "user",
"content": "The product of three prime numbers. There is a number that is the product of three prime factors whose sum of squares is equal to 2331. There are 7560 numbers (including 1) less than this number and coprime with it. The sum of all the divisors of this number (including 1 and th... | math | {
"style": "rule",
"ground_truth": "8987"
} | {
"split": "train",
"index": 8873,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the math scores of a certain high school approximately follow a normal distribution N(100, 100), calculate the percentage of students scoring between 80 and 120 points. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "95.44\\%"
} | {
"split": "train",
"index": 8874,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the function \\( f(x) = \\sin^4 \\left( \\frac{kx}{10} \\right) + \\cos^4 \\left( \\frac{kx}{10} \\right) \\), where \\( k \\) is a positive integer. If for any real number \\( a \\), the set \\(\\{ f(x) \\mid a < x < a+1 \\} = \\{ f(x) \\mid x \\in \\mathbf{R} \\}\\), then... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 8875,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of permutations of the 6 characters $a, b, c, d, e, f$ such that the subsequences $a c e$ and $d f$ do not appear. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "582"
} | {
"split": "train",
"index": 8876,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a $4\\times4$ grid where each row and each column forms an arithmetic sequence with four terms, find the value of $Y$, the center top-left square, with the first term of the first row being $3$ and the fourth term being $21$, and the first term of the fourth row being $15... | math | {
"style": "rule",
"ground_truth": "\\frac{43}{3}"
} | {
"split": "train",
"index": 8877,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the following seven numbers is exactly 19: $a_1 = 2.56$, $a_2 = 2.61$, $a_3 = 2.65$, $a_4 = 2.71$, $a_5 = 2.79$, $a_6 = 2.82$, $a_7 = 2.86$. It is desired to replace each $a_i$ by an integer approximation $A_i$, $1\\le i \\le 7$, so that the sum of the $A_i$'s is als... | math | {
"style": "rule",
"ground_truth": "61"
} | {
"split": "train",
"index": 8878,
"solution": "If any of the approximations $A_i$ is less than 2 or more than 3, the error associated with that term will be larger than 1, so the largest error will be larger than 1. However, if all of the $A_i$ are 2 or 3, the largest error will be less than 1. So in the best ca... |
deepscaler | [
{
"role": "user",
"content": "Let $A=(0,1),$ $B=(2,5),$ $C=(5,2),$ and $D=(7,0).$ A figure is created by connecting $A$ to $B,$ $B$ to $C,$ $C$ to $D,$ and $D$ to $A.$ The perimeter of $ABCD$ can be expressed in the form $a\\sqrt2+b\\sqrt{5}$ with $a$ and $b$ integers. What is the sum of $a$ and $b$? Let's ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 8879,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A mouse has a wheel of cheese which is cut into $2018$ slices. The mouse also has a $2019$ -sided die, with faces labeled $0,1,2,\\ldots, 2018$ , and with each face equally likely to come up. Every second, the mouse rolls the dice. If the dice lands on $k$ , and the mouse ... | math | {
"style": "rule",
"ground_truth": "2019"
} | {
"split": "train",
"index": 8880,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the value of $x$ for which $(2010 + x)^2 = 4x^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-670"
} | {
"split": "train",
"index": 8881,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that rectangle ABCD has dimensions AB = 7 and AD = 8, and right triangle DCE shares the same height as rectangle side DC = 7 and extends horizontally from D towards E, and the area of the right triangle DCE is 28, find the length of DE. Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "\\sqrt{113}"
} | {
"split": "train",
"index": 8882,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $PQR$, $PQ = 8$, $QR = 15$, and $PR = 17$. Point $S$ is the angle bisector of $\\angle QPR$. Find the length of $QS$ and then find the length of the altitude from $P$ to $QS$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 8883,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( z = \\frac{1+\\mathrm{i}}{\\sqrt{2}} \\). Then the value of \\( \\left(\\sum_{k=1}^{12} z^{k^{2}}\\right)\\left(\\sum_{k=1}^{12} \\frac{1}{z^{k^{2}}}\\right) \\) is ( ). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 8884,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be a $5$-digit number, and let $q$ and $r$ be the quotient and the remainder, respectively, when $n$ is divided by $100$. For how many values of $n$ is $q+r$ divisible by $11$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8181"
} | {
"split": "train",
"index": 8885,
"solution": "\n1. **Understanding the Problem:**\n Let $n$ be a 5-digit number. When $n$ is divided by 100, $q$ is the quotient and $r$ is the remainder. We need to find how many values of $n$ make $q+r$ divisible by 11.\n\n2. **Expressing $n$ in terms of $q$ and $r$:**\n Si... |
deepscaler | [
{
"role": "user",
"content": "What is the probability that when we roll four fair 6-sided dice, they won't all show the same number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{215}{216}"
} | {
"split": "train",
"index": 8886,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum to infinity of the terms of an infinite geometric progression is $6$. The sum of the first two terms is $4\\frac{1}{2}$. The first term of the progression is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9 or 3"
} | {
"split": "train",
"index": 8887,
"solution": "1. **Identify the given information and the formula to use:**\n - The sum to infinity of the terms of an infinite geometric progression is given as $6$.\n - The sum of the first two terms is $4\\frac{1}{2}$, which can be written as $\\frac{9}{2}$.\n - The sequ... |
deepscaler | [
{
"role": "user",
"content": "Calculate the total surface area of two hemispheres of radius 8 cm each, joined at their bases to form a complete sphere. Assume that one hemisphere is made of a reflective material that doubles the effective surface area for purposes of calculation. Express your answer in term... | math | {
"style": "rule",
"ground_truth": "384\\pi"
} | {
"split": "train",
"index": 8888,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\tan \\alpha = \\frac{1}{3}$ and $\\tan (\\alpha + \\beta) = \\frac{1}{2}$, find the value of $\\tan \\beta$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{7}"
} | {
"split": "train",
"index": 8889,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 8 sprinters in the Olympic 100-meter finals. Three of the sprinters are Americans. The gold medal goes to first place, silver to second, and bronze to third. In how many ways can the medals be awarded if at most one American gets a medal? Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 8890,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, there are more than three triangles. If each triangle has the same probability of being selected, what is the probability that a selected triangle has all or part of its interior shaded? Express your answer as a common fraction.\n\n[asy]\ndraw((0,0)--(1,0)--(0,1... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"split": "train",
"index": 8891,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the integer $n$, $-180 < n < 180$, such that $\\tan n^\\circ = \\tan 1500^\\circ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 8892,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the distinct non-zero numbers $x ( y - z),~ y(z - x),~ z(x - y )$ form a geometric progression with common ratio $r$, then $r$ satisfies the equation Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "r^2+r+1=0"
} | {
"split": "train",
"index": 8893,
"solution": "1. **Identify the terms of the geometric progression**: Given that $x(y-z)$, $y(z-x)$, and $z(x-y)$ form a geometric progression, we denote these terms as $a$, $ar$, and $ar^2$ respectively. Thus, we have:\n \\[ a = x(y-z), \\quad ar = y(z-x), \\quad ar^2 = z(x-y)... |
deepscaler | [
{
"role": "user",
"content": "(1) Simplify: $\\dfrac{\\sin(\\pi -\\alpha)\\cos(\\pi +\\alpha)\\sin(\\dfrac{\\pi}{2}+\\alpha)}{\\sin(-\\alpha)\\sin(\\dfrac{3\\pi}{2}+\\alpha)}$.\n\n(2) Given $\\alpha \\in (\\dfrac{\\pi}{2}, \\pi)$, and $\\sin(\\pi -\\alpha) + \\cos \\alpha = \\dfrac{7}{13}$, find $\\tan \\al... | math | {
"style": "rule",
"ground_truth": "-\\dfrac{12}{5}"
} | {
"split": "train",
"index": 8894,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the operation symbol \"△\" is defined as: $a \\triangle b = a + b + ab - 1$, and the operation symbol \"⊗\" is defined as: $a \\otimes b = a^2 - ab + b^2$, evaluate the value of $3 \\triangle (2 \\otimes 4)$. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 8895,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, $AB = 25 \\mbox{ cm},$ $AC = 20 \\mbox{ cm},$ and $\\angle A = 90^\\circ.$ What is the area of triangle $ABC?$\n\n[asy]\ndraw((0,0)--(25,0)--(0,20)--cycle,black+linewidth(1));\ndraw((0,0)--(1,0)--(1,1)--(0,1)--cycle,black+linewidth(1));\nlabel(\"$A$\",(0,0),SW);... | math | {
"style": "rule",
"ground_truth": "250"
} | {
"split": "train",
"index": 8896,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest possible distance between two points, one on the sphere of radius 15 with center $(3, -5, 7),$ and the other on the sphere of radius 95 with center $(-10, 20, -25)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "110 + \\sqrt{1818}"
} | {
"split": "train",
"index": 8897,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rectangle $ABCD$, $\\overline{AB}=20$ and $\\overline{BC}=10$. Let $E$ be a point on $\\overline{CD}$ such that $\\angle CBE=15^\\circ$. What is $\\overline{AE}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 8898,
"solution": "1. **Identify the given values and the problem requirements:**\n - Rectangle $ABCD$ has $\\overline{AB} = 20$ and $\\overline{BC} = 10$.\n - Point $E$ is on $\\overline{CD}$ such that $\\angle CBE = 15^\\circ$.\n - We need to find $\\overline{AE}$.\n\n2. **Use... |
deepscaler | [
{
"role": "user",
"content": "The sum of the first $20$ positive even integers is also the sum of four consecutive even integers. What is the largest of these four integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 8899,
"solution": ""
} |
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