data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Determine the range of $w(w + x)(w + y)(w + z)$, where $x, y, z$, and $w$ are real numbers such that\n\\[x + y + z + w = x^7 + y^7 + z^7 + w^7 = 0.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 32149,
"solution": "\nLet \\( x, y, z, \\) and \\( w \\) be real numbers such that they satisfy the equations:\n\n\\[\nx + y + z + w = 0\n\\]\n\\[\nx^7 + y^7 + z^7 + w^7 = 0.\n\\]\n\nWe are required to determine the range of the expression \\( (w + x)(w + y)(w + z)(w) \\).\n\nFirst, n... |
deepscaler | [
{
"role": "user",
"content": "Given $2^x = 8^{y+1}$ and $9^y = 3^{x-9}$, find the value of $x+y$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 32150,
"solution": "1. **Rewrite the equations with the same base:** \n Given $2^x = 8^{y+1}$, we know $8 = 2^3$, so:\n \\[\n 2^x = (2^3)^{y+1} = 2^{3(y+1)}\n \\]\n Similarly, given $9^y = 3^{x-9}$, and knowing $9 = 3^2$, we have:\n \\[\n 9^y = (3^2)^y = 3^{2y}\n \\]\... |
deepscaler | [
{
"role": "user",
"content": "It is possible to place positive integers into the vacant twenty-one squares of the $5 \\times 5$ square shown below so that the numbers in each row and column form arithmetic sequences. Find the number that must occupy the vacant square marked by the asterisk (*). Let's think ... | math | {
"style": "rule",
"ground_truth": "142"
} | {
"split": "train",
"index": 32151,
"solution": "First, let $a =$ the number to be placed in the first column, fourth row. Let $b =$ the number to be placed in the second column, fifth row. We can determine the entire first column and fifth row in terms of $a$ and $b$:\n$\\begin{tabular}[b]{|c|c|c|c|c|}\\hline 4a... |
deepscaler | [
{
"role": "user",
"content": "Let the random variable $\\xi$ follow the normal distribution $N(1, \\sigma^2)$ ($\\sigma > 0$). If $P(0 < \\xi < 1) = 0.4$, then find the value of $P(\\xi > 2)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.2"
} | {
"split": "train",
"index": 32152,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( f(n) = 3n^2 - 3n + 1 \\). Find the last four digits of \\( f(1) + f(2) + \\cdots + f(2010) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"split": "train",
"index": 32153,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $t_{n}$ equal the integer closest to $\\sqrt{n}$. What is the sum $\\frac{1}{t_{1}}+\\frac{1}{t_{2}}+\\frac{1}{t_{3}}+\\frac{1}{t_{4}}+\\cdots+\\frac{1}{t_{2008}}+\\frac{1}{t_{2009}}+\\frac{1}{t_{2010}}$? Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "88 \\frac{2}{3}"
} | {
"split": "train",
"index": 32154,
"solution": "First, we try a few values of $n$ to see if we can find a pattern in the values of $t_{n}$: So $t_{n}=1$ for 2 values of $n, 2$ for 4 values of $n, 3$ for 6 values of $n, 4$ for 8 values of $n$. We conjecture that $t_{n}=k$ for $2 k$ values of $n$. We will prove th... |
deepscaler | [
{
"role": "user",
"content": "Let $d_1 = a^2 + 3^a + a \\cdot 3^{(a+1)/2}$ and $d_2 = a^2 + 3^a - a \\cdot 3^{(a+1)/2}$. If $1 \\le a \\le 300$, how many integral values of $a$ are there such that $d_1 \\cdot d_2$ is a multiple of $7$? Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "43"
} | {
"split": "train",
"index": 32155,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of a triangle with angles $\\frac{1}{7} \\pi$ , $\\frac{2}{7} \\pi$ , and $\\frac{4}{7} \\pi $ , and radius of its circumscribed circle $R=1$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{7}}{4}"
} | {
"split": "train",
"index": 32156,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given sets \\( A = \\{1, 2, 3\\} \\) and \\( B = \\{4, 5, 6, 7\\} \\), with a mapping \\( f: A \\rightarrow B \\), if for \\( x \\in A \\), the expression \\( x + f(x) + x \\cdot f(x) \\) is odd, determine the number of such mappings \\( f \\). Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 32157,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( ABCD \\) be a trapezoid such that \\( (AB) \\) is parallel to \\( (CD) \\), \\( AB = 3 \\), \\( CD = 3 \\), \\( DA = 3 \\) and \\( \\widehat{ADC} = 120^\\circ \\). Determine the angle \\( \\widehat{CBA} \\) in degrees. Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 32158,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a hyperbola with asymptotes $2x \\pm y=0$, that passes through the intersection of the lines $x+y-3=0$ and $2x-y+3t=0$, where $-2 \\leq t \\leq 5$. Find the maximum possible length of the real axis of the hyperbola. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"split": "train",
"index": 32159,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $$Q(x) = \\left(\\frac{x^{24} - 1}{x - 1}\\right)^2 - x^{23}$$ and this polynomial has complex zeros in the form $z_k = r_k[\\cos(2\\pi\\alpha_k) + i\\sin(2\\pi\\alpha_k)]$, for $k = 1, 2, ..., 46$, where $0 < \\alpha_1 \\le \\alpha_2 \\le \\ldots \\le \\alpha_{46} < 1$ and... | math | {
"style": "rule",
"ground_truth": "\\frac{121}{575}"
} | {
"split": "train",
"index": 32160,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In right triangle $ABC$ the hypotenuse $\\overline{AB}=5$ and leg $\\overline{AC}=3$. The bisector of angle $A$ meets the opposite side in $A_1$. A second right triangle $PQR$ is then constructed with hypotenuse $\\overline{PQ}=A_1B$ and leg $\\overline{PR}=A_1C$. If the bisect... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{5}}{4}"
} | {
"split": "train",
"index": 32161,
"solution": "To solve this problem, we need to find the lengths of $A_1B$ and $A_1C$ first, and then use these to find $PP_1$.\n\n1. **Finding $BC$ in $\\triangle ABC$:**\n Since $\\triangle ABC$ is a right triangle with hypotenuse $AB = 5$ and leg $AC = 3$, we can find the o... |
deepscaler | [
{
"role": "user",
"content": "The store has 89 gold coins with numbers ranging from 1 to 89, each priced at 30 yuan. Among them, only one is a \"lucky coin.\" Feifei can ask an honest clerk if the number of the lucky coin is within a chosen subset of numbers. If the answer is \"Yes,\" she needs to pay a con... | math | {
"style": "rule",
"ground_truth": "130"
} | {
"split": "train",
"index": 32162,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiaoming started adding the page numbers of a book from page 1, sequentially until the end, and obtained a sum of 4979. Later, he discovered that one sheet (which includes two consecutive pages) was missing from the book. How many pages did the book originally have? Let's think... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 32163,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Perpendiculars $BE$ and $DF$ dropped from vertices $B$ and $D$ of parallelogram $ABCD$ onto sides $AD$ and $BC$, respectively, divide the parallelogram into three parts of equal area. A segment $DG$, equal to segment $BD$, is laid out on the extension of diagonal $BD$ beyond ve... | math | {
"style": "rule",
"ground_truth": "1:1"
} | {
"split": "train",
"index": 32164,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum distance between point $P$ on the line $y = 2x + 1$ and point $Q$ on the curve $y = x + \\ln{x}$. 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": 32165,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(C\\) be a cube with side length 4 and center \\(O\\). Let \\(S\\) be the sphere centered at \\(O\\) with radius 2. Let \\(A\\) be one of the vertices of the cube. Let \\(R\\) be the set of points in \\(C\\) but not in \\(S\\), which are closer to \\(A\\) than to any othe... | math | {
"style": "rule",
"ground_truth": "8 - \\frac{4\\pi}{3}"
} | {
"split": "train",
"index": 32166,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P, Q$ be functions defined as $P(x) = 3\\sqrt{x}$ and $Q(x) = x^2$. Calculate the value of $P(Q(P(Q(P(Q(5))))))$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "135"
} | {
"split": "train",
"index": 32167,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the area of the regular octagon $A B C D E F G H$ be $n$, and the area of the quadrilateral $A C E G$ be $m$. Calculate the value of $\\frac{m}{n}$. 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": 32168,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the line $y=ax+b$ is a tangent line of the graph of the function $f(x)=\\ln{x}-\\frac{1}{x}$, then the minimum value of $a+b$ is $\\_\\_\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 32169,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\cos \\frac {\\pi}{7}\\cos \\frac {2\\pi}{7}\\cos \\frac {4\\pi}{7}=$ ______. 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": 32170,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bot is held in orbit by the gravitational force of attraction to the planet, which creates a centripetal acceleration equal to $4 \\pi^{2}(2 R) / T^{2}$, where $R$ is the radius of the planet and $T$ is the period of revolution of the bot. From Newton's second law, we have:\n... | math | {
"style": "rule",
"ground_truth": "6000"
} | {
"split": "train",
"index": 32171,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify the expression $(-\\frac{1}{343})^{-2/3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 32172,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The coefficient of $x^2$ in the expansion of $(x-1) - (x-1)^2 + (x-1)^3 - (x-1)^4 + (x-1)^5$ is ____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-20"
} | {
"split": "train",
"index": 32173,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a regular tetrahedron \\( P-ABCD \\), where each face is an equilateral triangle with side length 1, points \\( M \\) and \\( N \\) are the midpoints of edges \\( AB \\) and \\( BC \\), respectively. Find the distance between the skew lines \\( MN \\) and \\( PC \\). Let's t... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{4}"
} | {
"split": "train",
"index": 32174,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all two-digit positive integers whose squares end with the digits 25? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "495"
} | {
"split": "train",
"index": 32175,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate\n\n$$\n\\operatorname{tg} \\frac{\\pi}{47} \\cdot \\operatorname{tg} \\frac{2 \\pi}{47} + \\operatorname{tg} \\frac{2 \\pi}{47} \\cdot \\operatorname{tg} \\frac{3 \\pi}{47} + \\ldots + \\operatorname{tg} \\frac{k \\pi}{47} \\cdot \\operatorname{tg} \\frac{(k+1) \\pi}{... | math | {
"style": "rule",
"ground_truth": "-2021"
} | {
"split": "train",
"index": 32176,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangles $ABC$ and $AFG$ have areas $3009$ and $9003,$ respectively, with $B=(0,0), C=(331,0), F=(800,450),$ and $G=(813,463).$ What is the sum of all possible $x$-coordinates of $A$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1400"
} | {
"split": "train",
"index": 32177,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A group of 40 boys and 28 girls stand hand in hand in a circle facing inwards. Exactly 18 of the boys give their right hand to a girl. How many boys give their left hand to a girl? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 32178,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point is chosen at random within a rectangle in the coordinate plane whose vertices are $(0, 0), (3030, 0), (3030, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\\tfrac{3}{4}$. Find the value of $d$ to the nearest tenth. Le... | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"split": "train",
"index": 32179,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of the expression \\( \\sqrt{\\frac{16^{12} + 8^{15}}{16^5 + 8^{16}}} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{2}}{4}"
} | {
"split": "train",
"index": 32180,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the fractional equation about $x$: $\\frac{x+m}{x+2}-\\frac{m}{x-2}=1$ has a solution not exceeding $6$, and the inequality system about $y$: $\\left\\{\\begin{array}{l}{m-6y>2}\\\\{y-4\\leq 3y+4}\\end{array}\\right.$ has exactly four integer solutions, then the sum of th... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 32181,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}=m \\overrightarrow{i}+5 \\overrightarrow{j}- \\overrightarrow{k}, \\overrightarrow{b}=3 \\overrightarrow{i}+ \\overrightarrow{j}+r \\overrightarrow{k}$, if $\\overrightarrow{a}//\\overrightarrow{b}$, find the values of real numbers $m=$ \\_\\_... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{5}"
} | {
"split": "train",
"index": 32182,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest number of consecutive non-negative integers whose sum is $120$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 32183,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph below shows the number of home runs in April for the top hitters in the league. What is the mean (average) number of home runs hit by these players?\n\n[asy]\ndraw((0,0)--(0,7)--(24,7)--(24,0)--cycle);\nlabel(\"KEY:\",(3,5));\nfill((3,2.5)..(3.5,2)..(3,1.5)..(2.5,2)..... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 32184,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Odell and Kershaw run for $30$ minutes on a circular track. Odell runs clockwise at $250 m/min$ and uses the inner lane with a radius of $50$ meters. Kershaw runs counterclockwise at $300 m/min$ and uses the outer lane with a radius of $60$ meters, starting on the same radial l... | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 32185,
"solution": "1. **Determine the Circumference of Each Track**:\n - Odell's track radius = $50$ meters, so the circumference is $C_O = 2\\pi \\times 50 = 100\\pi$ meters.\n - Kershaw's track radius = $60$ meters, so the circumference is $C_K = 2\\pi \\times 60 = 120\\pi$ met... |
deepscaler | [
{
"role": "user",
"content": "Line $l_1$ has equation $3x - 2y = 1$ and goes through $A = (-1, -2)$. Line $l_2$ has equation $y = 1$ and meets line $l_1$ at point $B$. Line $l_3$ has positive slope, goes through point $A$, and meets $l_2$ at point $C$. The area of $\\triangle ABC$ is $3$. What is the slope ... | math | {
"style": "rule",
"ground_truth": "\\tfrac34"
} | {
"split": "train",
"index": 32186,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a positive integer \\( n \\geq 3 \\), for an \\( n \\)-element real array \\(\\left(x_{1}, x_{2}, \\cdots, x_{n}\\right)\\), if every permutation \\( y_{1}, y_{2}, \\cdots, y_{n} \\) of it satisfies \\(\\sum_{i=1}^{n-1} y_{i} y_{i+1} \\geq -1\\), then the real array \\(\\... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 32187,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Integers $0 \\leq a, b, c, d \\leq 9$ satisfy $$\\begin{gathered} 6 a+9 b+3 c+d=88 \\\\ a-b+c-d=-6 \\\\ a-9 b+3 c-d=-46 \\end{gathered}$$ Find $1000 a+100 b+10 c+d$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6507"
} | {
"split": "train",
"index": 32188,
"solution": "Let $N=\\overline{a b c d}$ be an at most-four digit number. Note that the first equation gives us $N \\equiv 4(\\bmod 7)$. The second equation gives us $N \\equiv 6(\\bmod 11)$. The third equation gives us $N \\equiv 7$ $(\\bmod 13)$. Using CRT, we get $N \\equiv ... |
deepscaler | [
{
"role": "user",
"content": "What is the reciprocal of the common fraction form of $.\\overline{2}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac92"
} | {
"split": "train",
"index": 32189,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In our number system the base is ten. If the base were changed to four you would count as follows: \n$1,2,3,10,11,12,13,20,21,22,23,30,\\ldots$ The twentieth number would be: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "110"
} | {
"split": "train",
"index": 32190,
"solution": "To find the twentieth number in base 4, we need to convert the decimal number 20 into base 4.\n\n1. **Divide the number by 4 and record the remainder:**\n - $20 \\div 4 = 5$ with a remainder of $0$. This remainder is the least significant digit (rightmost digit) ... |
deepscaler | [
{
"role": "user",
"content": "Given the function f(x) = 2cos^2(x) - 2$\\sqrt{3}$sin(x)cos(x).\n(I) Find the monotonically decreasing interval of the function f(x);\n(II) Find the sum of all the real roots of the equation f(x) = $- \\frac{1}{3}$ in the interval [0, $\\frac{\\pi}{2}$]. Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"split": "train",
"index": 32191,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circles $C_1$ and $C_2$ are externally tangent, and they are both internally tangent to circle $C_3.$ The radii of $C_1$ and $C_2$ are 4 and 10, respectively, and the centers of the three circles are all collinear. A chord of $C_3$ is also a common external tangent of $C_1$ and... | math | {
"style": "rule",
"ground_truth": "405"
} | {
"split": "train",
"index": 32192,
"solution": "Call our desired length $x$. Note for any $X$ on $\\overline{AB}$ and $Y$ on $\\overline{O_1O_2}$ such that $\\overline{XY}\\perp\\overline{AB}$ that the function $f$ such that $f(\\overline{O_1Y})=\\overline{XY}$ is linear. Since $(0,4)$ and $(14,10)$, we can quic... |
deepscaler | [
{
"role": "user",
"content": "A $4\\times 4$ block of calendar dates is shown. First, the order of the numbers in the second and the fourth rows are reversed. Then, the numbers on each diagonal are added. What will be the positive difference between the two diagonal sums? Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 32193,
"solution": "1. **Identify the original diagonals**: \n - The main diagonal (from top left to bottom right) consists of the numbers: $1, 9, 17, 25$.\n - The secondary diagonal (from top right to bottom left) consists of the numbers: $4, 10, 16, 22$.\n\n2. **Calculate the or... |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate plane $(xOy)$, a point $A(2,0)$, a moving point $B$ on the curve $y= \\sqrt {1-x^{2}}$, and a point $C$ in the first quadrant form an isosceles right triangle $ABC$ with $\\angle A=90^{\\circ}$. The maximum length of the line segment $OC$ is _______.... | math | {
"style": "rule",
"ground_truth": "1+2 \\sqrt {2}"
} | {
"split": "train",
"index": 32194,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $f$ has the property that for each real number $x$ in its domain, $1/x$ is also in its domain and \\[\nf(x) + f\\left(\\frac{1}{x}\\right) = x.\n\\]What is the largest set of real numbers that can be in the domain of $f$?\n\n(a) ${\\{x\\mid x\\ne0\\}}$\n\n(b) ${\\{... | math | {
"style": "rule",
"ground_truth": "E"
} | {
"split": "train",
"index": 32195,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The operation \\( \\otimes \\) is defined by \\( a \\otimes b = \\frac{a}{b} + \\frac{b}{a} \\). What is the value of \\( 4 \\otimes 8 \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"split": "train",
"index": 32196,
"solution": "From the given definition, \\( 4 \\otimes 8 = \\frac{4}{8} + \\frac{8}{4} = \\frac{1}{2} + 2 = \\frac{5}{2} \\)."
} |
deepscaler | [
{
"role": "user",
"content": "Call a positive integer $N$ a 7-10 double if the digits of the base-$7$ representation of $N$ form a base-$10$ number that is twice $N$. For example, $51$ is a 7-10 double because its base-$7$ representation is $102$. What is the largest 7-10 double? Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "315"
} | {
"split": "train",
"index": 32197,
"solution": "Since this is an AIME problem, the maximum number of digits the 7-10 double can have is 3. Let the number be \\[abc\\] in base 7. Then the number in expanded form is \\[49a+7b+c\\] in base 7 and \\[100a+10b+c\\] in base 10. Since the number in base 7 is half the nu... |
deepscaler | [
{
"role": "user",
"content": "Al, Betty, and Clare split $\\$1000$ among them to be invested in different ways. Each begins with a different amount. At the end of one year they have a total of $\\$1500$. Betty and Clare have both doubled their money, whereas Al has managed to lose $\\$100$. What was Al's or... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 32198,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The mathematical giant Euler in history was the first to represent polynomials in terms of $x$ using the notation $f(x)$. For example, $f(x) = x^2 + 3x - 5$, and the value of the polynomial when $x$ equals a certain number is denoted by $f(\\text{certain number})$. For example,... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 32199,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the minimum distance between $(2019, 470)$ and $(21a - 19b, 19b + 21a)$ for $a, b \\in Z$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{101}"
} | {
"split": "train",
"index": 32200,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = x^3 - 3x^2 - 9x + 2$,\n(I) Determine the intervals of monotonicity for the function $f(x)$;\n(II) Find the minimum value of function $f(x)$ on the interval [-2, 2]. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-20"
} | {
"split": "train",
"index": 32201,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a $9 \\times 9$ grid of squares. Haruki fills each square in this grid with an integer between 1 and 9 , inclusive. The grid is called a super-sudoku if each of the following three conditions hold: - Each column in the grid contains each of the numbers $1,2,3,4,5,6,7,8... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 32202,
"solution": "Without loss of generality, suppose that the top left corner contains a 1 , and examine the top left $3 \\times 4$ : \\begin{tabular}{|c|c|c|c|} \\hline 1 & x & x & x \\\\ \\hline x & x & x & $*$ \\\\ \\hline x & x & x & $*$ \\\\ \\hline \\end{tabular} There cannot... |
deepscaler | [
{
"role": "user",
"content": "Given that \\( O \\) is the circumcenter of \\(\\triangle ABC\\), and \\( 3 \\overrightarrow{OA} + 4 \\overrightarrow{OB} + 5 \\overrightarrow{OC} = \\overrightarrow{0} \\), find the value of \\( \\cos \\angle BAC \\). Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{10}"
} | {
"split": "train",
"index": 32203,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $c \\geq 4$ be an even integer. In some football league, each team has a home uniform and anaway uniform. Every home uniform is coloured in two different colours, and every away uniformis coloured in one colour. A team’s away uniform cannot be coloured in one of the colours... | math | {
"style": "rule",
"ground_truth": "c\\lfloor\\frac{c^2}4\\rfloor"
} | {
"split": "train",
"index": 32204,
"solution": "To solve this problem, we need to determine the maximum number of teams in a football league under the given constraints. Each team has a home uniform with two distinct colors and an away uniform with a single color. There are at most \\( c \\) distinct colors avai... |
deepscaler | [
{
"role": "user",
"content": "Find all the roots of the polynomial $x^{5}-5 x^{4}+11 x^{3}-13 x^{2}+9 x-3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1, \\frac{3+\\sqrt{3} i}{2}, \\frac{1-\\sqrt{3} i}{2}, \\frac{3-\\sqrt{3} i}{2}, \\frac{1+\\sqrt{3} i}{2}"
} | {
"split": "train",
"index": 32205,
"solution": "The $x^{5}-5 x^{4}$ at the beginning of the polynomial motivates us to write it as $(x-1)^{5}+x^{3}-3 x^{2}+4 x-2$ and again the presence of the $x^{3}-3 x^{2}$ motivates writing the polynomial in the form $(x-1)^{5}+(x-1)^{3}+(x-1)$. Let $a$ and $b$ be the roots o... |
deepscaler | [
{
"role": "user",
"content": "What is the value of $x + y$ if the sequence $2, ~6, ~10, \\ldots, ~x, ~y, ~26$ is an arithmetic sequence? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 32206,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of the power function $f(x)$ passes through the point $(3, \\frac{1}{9})$, find the maximum value of the function $g(x) = (x-1)f(x)$ on the interval $[1,3]$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 32207,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "40 pikes were released into a pond. A pike is considered well-fed if it has eaten three other pikes (whether well-fed or hungry). What is the maximum number of pikes that can be well-fed? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 32208,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\frac{a}{10^x-1}+\\frac{b}{10^x+2}=\\frac{2 \\cdot 10^x+3}{(10^x-1)(10^x+2)}$ is an identity for positive rational values of $x$, then the value of $a-b$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 32209,
"solution": "Given the equation:\n\\[\n\\frac{a}{10^x-1}+\\frac{b}{10^x+2}=\\frac{2 \\cdot 10^x+3}{(10^x-1)(10^x+2)}\n\\]\nLet's simplify by letting \\( y = 10^x \\). The equation becomes:\n\\[\n\\frac{a}{y-1}+\\frac{b}{y+2}=\\frac{2y+3}{(y-1)(y+2)}\n\\]\nMultiplying each term ... |
deepscaler | [
{
"role": "user",
"content": "In right triangle $DEF$, $DE=15$, $DF=9$ and $EF=12$ units. What is the distance from $F$ to the midpoint of segment $DE$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7.5"
} | {
"split": "train",
"index": 32210,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular tetrahedron with four equilateral triangular faces has a sphere inscribed within it and another sphere circumscribed about it. Each of the four faces of the tetrahedron is tangent to a unique external sphere which is also tangent to the circumscribed sphere, but now t... | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"split": "train",
"index": 32211,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram below, $ABCD$ is a trapezoid such that $\\overline{AB}\\parallel \\overline{CD}$ and $\\overline{AC}\\perp\\overline{CD}$. If $CD = 15$, $\\tan D = 2$, and $\\tan B = 3$, then what is $BC$? Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "10\\sqrt{10}"
} | {
"split": "train",
"index": 32212,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express the following as a common fraction: $\\sqrt[3]{4\\div 13.5}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac23"
} | {
"split": "train",
"index": 32213,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a table tennis match between player A and player B, the \"best of five sets\" rule is applied, which means the first player to win three sets wins the match. If the probability of player A winning a set is $\\dfrac{2}{3}$, and the probability of player B winning a set is $\\... | math | {
"style": "rule",
"ground_truth": "\\dfrac{8}{27}"
} | {
"split": "train",
"index": 32214,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Emily sees a ship traveling at a constant speed along a straight section of a river. She walks parallel to the riverbank at a uniform rate faster than the ship. She counts $210$ equal steps walking from the back of the ship to the front. Walking in the opposite direction, she c... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 32215,
"solution": "1. **Define Variables:**\n Let $L$ be the length of the ship, $E$ be the length of Emily's step, and $S$ be the length of the ship's step (the distance the ship travels while Emily takes one step).\n\n2. **Set Up Equations:**\n - When Emily walks from the back ... |
deepscaler | [
{
"role": "user",
"content": "If $A=20^{\\circ}$ and $B=25^{\\circ}$, then the value of $(1+\\tan A)(1+\\tan B)$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 32216,
"solution": "\n1. **Expanding the Expression**:\n \\[\n (1+\\tan A)(1+\\tan B) = 1 + \\tan A + \\tan B + \\tan A \\tan B\n \\]\n\n2. **Using the Angle Sum Identity for Tangent**:\n Since $A + B = 45^\\circ$, we know that $\\tan(45^\\circ) = 1$. Using the tangent sum for... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sin(2x- \\frac{\\pi}{6})$, determine the horizontal shift required to obtain the graph of the function $g(x)=\\sin(2x)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{12}"
} | {
"split": "train",
"index": 32217,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ and $y$ be nonzero real numbers. Let $m$ and $M$ be the minimium value and maximum value of\n\\[\\frac{|x + y|}{|x| + |y|},\\]respectively. Find $M - m.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 32218,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a general triangle \\(ABC\\) with points \\(K, L, M, N, U\\) on its sides:\n\n- Point \\(K\\) is the midpoint of side \\(AC\\).\n- Point \\(U\\) is the midpoint of side \\(BC\\).\n- Points \\(L\\) and \\(M\\) lie on segments \\(CK\\) and \\(CU\\) respectively, such that \... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 32219,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The 64 whole numbers from 1 through 64 are written, one per square, on a checkerboard (an 8 by 8 array of 64 squares). The first 8 numbers are written in order across the first row, the next 8 across the second row, and so on. After all 64 numbers are written, the sum of the ... | math | {
"style": "rule",
"ground_truth": "130"
} | {
"split": "train",
"index": 32220,
"solution": "1. **Identify the positions of the corner numbers on the checkerboard:**\n - The top left corner is the first square, so the number is $1$.\n - The top right corner is the last square of the first row, which is the 8th square, so the number is $8$.\n - The bo... |
deepscaler | [
{
"role": "user",
"content": "A permutation of a finite set is a one-to-one function from the set to itself; for instance, one permutation of $\\{1,2,3,4\\}$ is the function $\\pi$ defined such that $\\pi(1)=1, \\pi(2)=3$, $\\pi(3)=4$, and $\\pi(4)=2$. How many permutations $\\pi$ of the set $\\{1,2, \\ldot... | math | {
"style": "rule",
"ground_truth": "945"
} | {
"split": "train",
"index": 32221,
"solution": "For each such $\\pi$, the elements of $\\{1,2, \\ldots, 10\\}$ can be arranged into pairs $\\{i, j\\}$ such that $\\pi(i)=j ; \\pi(j)=i$. Choosing a permutation $\\pi$ is thus tantamount to choosing a partition of $\\{1,2, \\ldots, 10\\}$ into five disjoint pairs. ... |
deepscaler | [
{
"role": "user",
"content": "There are 10 sprinters in the Olympic 100-meter finals. Four of the sprinters are from Spain. The gold, silver, and bronze medals are awarded to the top three finishers. In how many ways can the medals be awarded if at most two Spaniards get medals? Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "696"
} | {
"split": "train",
"index": 32222,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a bus station in the city, there are 10 waiting seats arranged in a row. Now, if 4 passengers randomly choose some seats to wait, the number of ways to arrange them so that there are exactly 5 consecutive empty seats is $\\boxed{480}$. Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "480"
} | {
"split": "train",
"index": 32223,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $XYZ$, $XY = 4$, $XZ = 3$, and $YZ = 5$. The medians $XM$, $YN$, and $ZL$ of triangle $XYZ$ intersect at the centroid $G$. Let the projections of $G$ onto $YZ$, $XZ$, and $XY$ be $P$, $Q$, and $R$, respectively. Find $GP + GQ + GR$. Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "\\frac{47}{15}"
} | {
"split": "train",
"index": 32224,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square is completely covered by a large circle and each corner of the square touches a smaller circle of radius \\( r \\). The side length of the square is 6 units. What is the radius \\( R \\) of the large circle? Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2}"
} | {
"split": "train",
"index": 32225,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiaoming saw a tractor pulling a rope slowly on the road. Xiaoming decided to measure the length of the rope. If Xiaoming walks in the same direction as the tractor, it takes 140 steps to walk from one end of the rope to the other end; if Xiaoming walks in the opposite directio... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 32226,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Perform the calculations. \n$(54+38) \\times 15$ \n$1500-32 \\times 45$ \n$157 \\times (70 \\div 35)$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "314"
} | {
"split": "train",
"index": 32227,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle ABC$ be a right triangle such that $B$ is a right angle. A circle with diameter of $BC$ meets side $AC$ at $D.$ If $AD = 1$ and $BD = 4,$ then what is $CD$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 32228,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five packages are delivered to five houses, one to each house. If these packages are randomly delivered, what is the probability that exactly three of them are delivered to the correct houses? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"split": "train",
"index": 32229,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a positive integer \\( n \\), if there exist positive integers \\( a \\) and \\( b \\) such that \\( n = a + b + a \\times b \\), then \\( n \\) is called a \"good number\". For example, \\( 3 = 1 + 1 + 1 \\times 1 \\), so 3 is a \"good number\". Among the 100 positive inte... | math | {
"style": "rule",
"ground_truth": "74"
} | {
"split": "train",
"index": 32230,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If a positive integer \\( N \\) can be expressed as \\( \\lfloor x \\rfloor + \\lfloor 2x \\rfloor + \\lfloor 3x \\rfloor \\) for some real number \\( x \\), then we say that \\( N \\) is \"visible\"; otherwise, we say that \\( N \\) is \"invisible\". For example, 8 is visible ... | math | {
"style": "rule",
"ground_truth": "6034"
} | {
"split": "train",
"index": 32231,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that line l: x - y + 1 = 0 is tangent to the parabola C with focus F and equation y² = 2px (p > 0).\n(I) Find the equation of the parabola C;\n(II) The line m passing through point F intersects parabola C at points A and B. Find the minimum value of the sum of the distanc... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 32232,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given sets $A=\\{x,\\frac{y}{x},1\\}$ and $B=\\{{x}^{2},x+y,0\\}$, if $A=B$, then $x^{2023}+y^{2024}=\\_\\_\\_\\_\\_\\_.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 32233,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Which of the following is equal to $9^{4}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3^{8}"
} | {
"split": "train",
"index": 32234,
"solution": "Since $9=3 \\times 3$, then $9^{4}=(3 \\times 3)^{4}=3^{4} \\times 3^{4}=3^{8}$. Alternatively, we can note that $9^{4}=9 \\times 9 \\times 9 \\times 9=(3 \\times 3) \\times(3 \\times 3) \\times(3 \\times 3) \\times(3 \\times 3)=3^{8}$."
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ be a positive real number such that $x + \\frac{1}{x} = 98.$ Find\\[\\sqrt{x} + \\frac{1}{\\sqrt{x}}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 32235,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given integer $n\\geq 2$. Find the minimum value of $\\lambda {}$, satisfy that for any real numbers $a_1$, $a_2$, $\\cdots$, ${a_n}$ and ${b}$,\n$$\\lambda\\sum\\limits_{i=1}^n\\sqrt{|a_i-b|}+\\sqrt{n\\left|\\sum\\limits_{i=1}^na_i\\right|}\\geqslant\\sum\\limits_{i=1}^n\\sqrt... | math | {
"style": "rule",
"ground_truth": "\\frac{n-1 + \\sqrt{n-1}}{\\sqrt{n}}"
} | {
"split": "train",
"index": 32236,
"solution": "\nGiven an integer \\( n \\geq 2 \\), we aim to find the minimum value of \\( \\lambda \\) such that for any real numbers \\( a_1, a_2, \\ldots, a_n \\) and \\( b \\), the following inequality holds:\n\\[\n\\lambda \\sum_{i=1}^n \\sqrt{|a_i - b|} + \\sqrt{n \\left|... |
deepscaler | [
{
"role": "user",
"content": "The first 14 terms of the sequence $\\{a_n\\}$ are 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38. According to this pattern, find $a_{16}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 32237,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the positive value of $x$ which satisfies\n\\[\\log_5 (x - 2) + \\log_{\\sqrt{5}} (x^3 - 2) + \\log_{\\frac{1}{5}} (x - 2) = 4.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 32238,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a$, $b$, $c$, $d$, $e$, and $f$ are integers for which $512x^3 + 125 = (ax^2 + bx + c)(dx^2 + ex + f)$ for all $x$, then what is $a^2+b^2+c^2+d^2+e^2+f^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6410"
} | {
"split": "train",
"index": 32239,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two congruent circles centered at points $A$ and $B$ each pass through the other circle's center. The line containing both $A$ and $B$ is extended to intersect the circles at points $C$ and $D$. The circles intersect at two points, one of which is $E$. What is the degree measur... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 32240,
"solution": "\n1. **Identify the Shape of $\\triangle EAB$**:\n Since the circles are congruent and each circle's center lies on the other circle, the distance between $A$ and $B$ equals the radius of the circles. Thus, $EA = EB = AB$, making $\\triangle EAB$ equilateral.\n\n... |
deepscaler | [
{
"role": "user",
"content": "For the fractional equation involving $x$, $\\frac{x+m}{x-2}+\\frac{1}{2-x}=3$, if it has a root with an increase, then $m=\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 32241,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a$ is an element randomly selected from the set $\\{1, 2, 3, 4, 5, 6, 7\\}$, then the probability that the circle $C: x^2 + (y-2)^2 = 1$ is contained inside the circle $O: x^2 + y^2 = a^2$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{7}"
} | {
"split": "train",
"index": 32242,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $AB= 400$, $BC=480$, and $AC=560$. An interior point $P$ is identified, and segments are drawn through $P$ parallel to the sides of the triangle. These three segments are of equal length $d$. Determine $d$. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "218\\frac{2}{9}"
} | {
"split": "train",
"index": 32243,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number $n$ can be written in base $14$ as $\\underline{a}\\text{ }\\underline{b}\\text{ }\\underline{c}$, can be written in base $15$ as $\\underline{a}\\text{ }\\underline{c}\\text{ }\\underline{b}$, and can be written in base $6$ as $\\underline{a}\\text{ }\\underline{c}\... | math | {
"style": "rule",
"ground_truth": "925"
} | {
"split": "train",
"index": 32244,
"solution": "We have these equations: $196a+14b+c=225a+15c+b=222a+37c$. Taking the last two we get $3a+b=22c$. Because $c \\neq 0$ otherwise $a \\ngtr 0$, and $a \\leq 5$, $c=1$.\n Then we know $3a+b=22$. Taking the first two equations we see that $29a+14c=13b$. Combining the t... |
deepscaler | [
{
"role": "user",
"content": "Let $S_{n}$ be the sum of the first $n$ terms of an arithmetic sequence $\\{a_{n}\\}$ with a common difference of $d$. If $a_{1}=190$, $S_{20} \\gt 0$, and $S_{24} \\lt 0$, then one possible value for the integer $d$ is ______. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "-17"
} | {
"split": "train",
"index": 32245,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three concentric circles with radii 5 meters, 10 meters, and 15 meters, form the paths along which an ant travels moving from one point to another symmetrically. The ant starts at a point on the smallest circle, moves radially outward to the third circle, follows a path on each... | math | {
"style": "rule",
"ground_truth": "\\frac{65\\pi}{3} + 20"
} | {
"split": "train",
"index": 32246,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ahn chooses a two-digit integer, subtracts it from 200, and doubles the result. What is the largest number Ahn can get? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "380"
} | {
"split": "train",
"index": 32247,
"solution": "1. **Identify the operation**: Ahn performs the operation $2(200 - n)$, where $n$ is a two-digit integer.\n\n2. **Determine the range of $n$**: Since $n$ is a two-digit integer, $10 \\leq n \\leq 99$.\n\n3. **Maximize the expression**: To maximize $2(200 - n)$, we ... |
deepscaler | [
{
"role": "user",
"content": "It is now 3:00:00 PM, as read on a 12-hour digital clock. In 315 hours, 58 minutes, and 16 seconds, the time will be $X:Y:Z$. What is the value of $X + Y + Z$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "77"
} | {
"split": "train",
"index": 32248,
"solution": ""
} |
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