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 function $f(x)$ that satisfies: (1) For any positive real number $x$, it always holds that $f(2x) = 2f(x)$; (2) When $x \\in (1, 2)$, $f(x) = 2 - x$. If $f(a) = f(2020)$, determine the smallest positive real number $a$. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 37849,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "No two students in Mrs. Vale's 26-student mathematics class have the same two initials. Each student's first name and last name begin with the same letter. If the letter ``Y'' is considered a vowel, what is the probability of randomly picking a student whose initials are vowels... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{13}"
} | {
"split": "train",
"index": 37850,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $y=\\cos({2x+\\frac{π}{3}})$, determine the horizontal shift of the graph of the function $y=\\sin 2x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{12}"
} | {
"split": "train",
"index": 37851,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Quantities $a$ and $b$ vary inversely. When $a$ is $800$, $b$ is $0.5$. If the product of $a$ and $b$ increases by $200$ when $a$ is doubled, what is $b$ when $a$ is $1600$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.375"
} | {
"split": "train",
"index": 37852,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A solid triangular prism is made up of 27 identical smaller solid triangular prisms. The length of every edge of each of the smaller prisms is 1. If the entire outer surface of the larger prism is painted, what fraction of the total surface area of all the smaller prisms is pai... | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"split": "train",
"index": 37853,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The point $(1,1,1)$ is rotated $180^\\circ$ about the $y$-axis, then reflected through the $yz$-plane, reflected through the $xz$-plane, rotated $180^\\circ$ about the $y$-axis, and reflected through the $xz$-plane. Find the coordinates of the point now. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "(-1,1,1)"
} | {
"split": "train",
"index": 37854,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The store owner bought 2000 pens at $0.15 each and plans to sell them at $0.30 each, calculate the number of pens he needs to sell to make a profit of exactly $150. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"split": "train",
"index": 37855,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = (\\sin x + \\cos x)^2 + \\cos 2x - 1$.\n(1) Find the smallest positive period of the function $f(x)$;\n(2) Find the maximum and minimum values of $f(x)$ in the interval $\\left[-\\frac{\\pi}{4}, \\frac{\\pi}{4}\\right]$. Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "-\\sqrt{2}"
} | {
"split": "train",
"index": 37856,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two vertices of a cube are given in space. The locus of points that could be a third vertex of the cube is the union of $n$ circles. Find $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 37857,
"solution": "Let the distance between the two given vertices be 1. If the two given vertices are adjacent, then the other vertices lie on four circles, two of radius 1 and two of radius $\\sqrt{2}$. If the two vertices are separated by a diagonal of a face of the cube, then the... |
deepscaler | [
{
"role": "user",
"content": "Given functions $f(x)=xe^x$ and $g(x)=-\\frac{lnx}{x}$, if $f(x_{1})=g(x_{2})=t\\left( \\gt 0\\right)$, find the maximum value of $\\frac{{x}_{1}}{{x}_{2}{e}^{t}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{e}"
} | {
"split": "train",
"index": 37858,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of the definite integral $\\int_{0}^{1} ( \\sqrt{1-(x-1)^{2}}-{x}^{2})dx$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4} - \\frac{1}{3}"
} | {
"split": "train",
"index": 37859,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A freight train travels 1 mile in 1 minute 30 seconds. At this rate, how many miles will the train travel in 1 hour? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 37860,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Real numbers \\( x_{1}, x_{2}, \\cdots, x_{2001} \\) satisfy \\( \\sum_{k=1}^{2000} \\left| x_{k} - x_{k+1} \\right| = 2001 \\). Let \\( y_{k} = \\frac{1}{k} \\left( x_{1} + x_{2} + \\cdots + x_{k} \\right) \\) for \\( k = 1, 2, \\cdots, 2001 \\). Find the maximum possible valu... | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"split": "train",
"index": 37861,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all real numbers $x$ satisfying the equation $x^{3}-8=16 \\sqrt[3]{x+1}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2,1 \\pm \\sqrt{5}"
} | {
"split": "train",
"index": 37862,
"solution": "Let $f(x)=\\frac{x^{3}-8}{8}$. Then $f^{-1}(x)=\\sqrt[3]{8x+8}=2\\sqrt[3]{x+1}$, and so the given equation is equivalent to $f(x)=f^{-1}(x)$. This implies $f(f(x))=x$. However, as $f$ is monotonically increasing, this implies that $f(x)=x$. As a result, we have $\\... |
deepscaler | [
{
"role": "user",
"content": "Consider a unit cube in a coordinate system with vertices $A(0,0,0)$, $A'(1,1,1)$, and other vertices placed accordingly. A regular octahedron has vertices placed at fractions $\\frac{1}{3}$ and $\\frac{2}{3}$ along the segments connecting $A$ with $A'$'s adjacent vertices and ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{3}"
} | {
"split": "train",
"index": 37863,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The two squares shown share the same center $O$ and have sides of length 1. The length of $\\overline{AB}$ is $43/99$ and the area of octagon $ABCDEFGH$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$\n[asy] //code taken from thread for problem r... | math | {
"style": "rule",
"ground_truth": "185"
} | {
"split": "train",
"index": 37864,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If point P is one of the intersections of the hyperbola with foci A(-√10,0), B(√10,0) and a real axis length of 2√2, and the circle x^2 + y^2 = 10, calculate the value of |PA| + |PB|. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6\\sqrt{2}"
} | {
"split": "train",
"index": 37865,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse E: $$\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$$ ($$a > b > 0$$) passing through point Q ($$\\frac{\\sqrt{2}}{2}, \\frac{\\sqrt{3}}{2}$$), the product of the slopes of the lines connecting the moving point P on the ellipse to the two endpoints of the minor axis i... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 37866,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $A, B, C$, and $D$ are four circles of radius $r>0$ centered about the points $(0, r),(r, 0)$, $(0,-r)$, and $(-r, 0)$ in the plane. Let $O$ be a circle centered at $(0,0)$ with radius $2 r$. In terms of $r$, what is the area of the union of circles $A, B, C$, and $D$ s... | math | {
"style": "rule",
"ground_truth": "8 r^{2}"
} | {
"split": "train",
"index": 37867,
"solution": "Let $U$ denote the union of the four circles, so we seek $$U-([O]-U)=2 U-[O]=2\\left[(2 r)^{2}+4 \\cdot \\frac{1}{2} \\pi r^{2}\\right]-\\pi(2 r)^{2}=8 r^{2}$$ (Here we decompose $U$ into the square $S$ with vertices at $( \\pm r, \\pm r)$ and the four semicircular... |
deepscaler | [
{
"role": "user",
"content": "Given that acute angles $\\alpha$ and $\\beta$ satisfy $\\alpha+2\\beta=\\frac{2\\pi}{3}$ and $\\tan\\frac{\\alpha}{2}\\tan\\beta=2-\\sqrt{3}$, find the value of $\\alpha +\\beta$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{12}"
} | {
"split": "train",
"index": 37868,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Tetrahedron $ABCD$ has $AB=5$, $AC=3$, $BC=4$, $BD=4$, $AD=3$, and $CD=\\frac{12}5\\sqrt2$. What is the volume of the tetrahedron? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{24}5"
} | {
"split": "train",
"index": 37869,
"solution": "1. **Identify the midpoint and compute segment lengths**: Let the midpoint of $CD$ be $E$. Since $CD = \\frac{12}{5}\\sqrt{2}$, we have $CE = \\frac{1}{2}CD = \\frac{6}{5}\\sqrt{2}$.\n\n2. **Use the Pythagorean Theorem in $\\triangle ACE$ and $\\triangle BCE$**:\n ... |
deepscaler | [
{
"role": "user",
"content": "$R$ varies directly as $S$ and inversely as $T$. When $R = \\frac{4}{3}$ and $T = \\frac{9}{14}$, $S = \\frac{3}{7}$. Find $S$ when $R = \\sqrt{48}$ and $T = \\sqrt{75}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 37870,
"solution": "1. **Establish the relationship**: Given that $R$ varies directly as $S$ and inversely as $T$, we can express this relationship mathematically as:\n \\[\n R = c \\cdot \\frac{S}{T}\n \\]\n where $c$ is a constant.\n\n2. **Determine the constant $c$**: Using... |
deepscaler | [
{
"role": "user",
"content": "Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each other in the same row, and no... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 37871,
"solution": "1. **Identify the constraints**: Each row has three seats, and siblings must not sit next to each other in the same row or directly in front of each other. This implies that each sibling pair must be split between the two rows.\n\n2. **Assign siblings to rows**: Le... |
deepscaler | [
{
"role": "user",
"content": "Al-Karhi's rule for approximating the square root. If \\(a^{2}\\) is the largest square contained in the given number \\(N\\), and \\(r\\) is the remainder, then\n\n$$\n\\sqrt{N}=\\sqrt{a^{2}+r}=a+\\frac{r}{2a+1}, \\text{ if } r<2a+1\n$$\n\nExplain how Al-Karhi might have deriv... | math | {
"style": "rule",
"ground_truth": "20.366"
} | {
"split": "train",
"index": 37872,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle with side lengths in the ratio 2:3:4 is inscribed in a circle of radius 4. What is the area of the triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{15}"
} | {
"split": "train",
"index": 37873,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(ABC\\) be a triangle with \\(AB=7\\), \\(BC=9\\), and \\(CA=4\\). Let \\(D\\) be the point such that \\(AB \\parallel CD\\) and \\(CA \\parallel BD\\). Let \\(R\\) be a point within triangle \\(BCD\\). Lines \\(\\ell\\) and \\(m\\) going through \\(R\\) are parallel to \... | math | {
"style": "rule",
"ground_truth": "180"
} | {
"split": "train",
"index": 37874,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given real numbers $x$ and $y$ that satisfy the equation $x^{2}+y^{2}-4x+6y+12=0$, find the minimum value of $|2x-y-2|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5-\\sqrt{5}"
} | {
"split": "train",
"index": 37875,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The price of Type A remote control car is 46.5 yuan, and the price of Type B remote control car is 54.5 yuan. Lele has 120 yuan. If he buys both types of remote control cars, will he have enough money? If so, how much money will he have left after the purchase? Let's think step... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 37876,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the function $f(x)$ satisfies: $4f(x)f(y)=f(x+y)+f(x-y)$ $(x,y∈R)$ and $f(1)= \\frac{1}{4}$, find $f(2014)$. 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": 37877,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system on a plane, the parametric equations of curve $C$ are given by $\\begin{cases} x=5\\cos \\alpha \\\\ y=\\sin \\alpha \\end{cases}$ where $\\alpha$ is a parameter, and point $P$ has coordinates $(3 \\sqrt {2},0)$.\n(1) Determine the shape of ... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{13}"
} | {
"split": "train",
"index": 37878,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the ratio of the area of square $WXYZ$ to the area of square $PQRS$ if $PQRS$ has side length 2 and $W, X, Y, Z$ are the midpoints of the sides of $PQRS$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1: 2"
} | {
"split": "train",
"index": 37879,
"solution": "Since square $PQRS$ has side length 2, then $PQ=QR=RS=SP=2$. Since $W, X, Y, Z$ are the midpoints of the sides of $PQRS$, then $PW=PZ=1$. Since $\\angle ZPW=90^{\\circ}$, then $WZ=\\sqrt{PW^{2}+PZ^{2}}=\\sqrt{1^{2}+1^{2}}=\\sqrt{2}$. Therefore, square $WXYZ$ has si... |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be a positive integer, and let $b_0, b_1, \\dots, b_n$ be a sequence of real numbers such that $b_0 = 25$, $b_1 = 56$, $b_n = 0$, and\n$$ b_{k+1} = b_{k-1} - \\frac{7}{b_k} $$\nfor $k = 1, 2, \\dots, n-1$. Find $n$. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "201"
} | {
"split": "train",
"index": 37880,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the numbers from the set $\\{1, 2, 3, \\ldots, 100\\}$ have a perfect square factor other than one? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 37881,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose two distinct integers are chosen from between 5 and 17, inclusive. What is the probability that their product is odd? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{7}{26}"
} | {
"split": "train",
"index": 37882,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\sin\\left(\\frac{\\pi}{3}+\\frac{\\alpha}{6}\\right)=-\\frac{3}{5}$, $\\cos\\left(\\frac{\\pi}{12}-\\frac{\\beta}{2}\\right)=-\\frac{12}{13}$, $-5\\pi < \\alpha < -2\\pi$, $-\\frac{11\\pi}{6} < \\beta < \\frac{\\pi}{6}$,\n\nFind the value of $\\sin \\left(\\frac{\\alph... | math | {
"style": "rule",
"ground_truth": "\\frac{16}{65}"
} | {
"split": "train",
"index": 37883,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two arithmetic sequences $\\{a_n\\}$ and $\\{b_n\\}$, the sums of the first $n$ terms are denoted as $S_n$ and $T_n$ respectively. If for any natural number $n$, it holds that $\\dfrac{S_n}{T_n} = \\dfrac{2n-3}{4n-3}$, calculate the value of $\\dfrac{a_3+a_{15}}{2(b_3+b_9... | math | {
"style": "rule",
"ground_truth": "\\dfrac{19}{41}"
} | {
"split": "train",
"index": 37884,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At Easter-Egg Academy, each student has two eyes, each of which can be eggshell, cream, or cornsilk. It is known that $30 \\%$ of the students have at least one eggshell eye, $40 \\%$ of the students have at least one cream eye, and $50 \\%$ of the students have at least one co... | math | {
"style": "rule",
"ground_truth": "80 \\%"
} | {
"split": "train",
"index": 37885,
"solution": "For the purposes of this solution, we abbreviate \"eggshell\" by \"egg\", and \"cornsilk\" by \"corn\". We know that there are only six combinations of eye color possible: egg-cream, egg-corn, egg-egg, cream-corn, cream-cream, corn-corn. If we let the proportions f... |
deepscaler | [
{
"role": "user",
"content": "If\n\\[\\frac{\\sin x}{\\cos y} + \\frac{\\sin y}{\\cos x} = 1 \\quad \\text{and} \\quad \\frac{\\cos x}{\\sin y} + \\frac{\\cos y}{\\sin x} = 6,\\]then find $\\frac{\\tan x}{\\tan y} + \\frac{\\tan y}{\\tan x}.$ Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "\\frac{124}{13}"
} | {
"split": "train",
"index": 37886,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Selected Exercise $4-4$: Coordinate Systems and Parametric Equations\n\nIn the rectangular coordinate system $xOy$, the parametric equations of the curve $C$ are given by $\\begin{cases} & x=\\cos \\theta \\\\ & y=\\sin \\theta \\end{cases}$, where $\\theta$ is the parameter. I... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{2}}{2} + 1"
} | {
"split": "train",
"index": 37887,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of all nonzero real numbers. The function $f : S \\to S$ satisfies the following two properties:\n\n(i) First,\n\\[f \\left( \\frac{1}{x} \\right) = xf(x)\\]for all $x \\in S.$\n\n(ii) Second,\n\\[f \\left( \\frac{1}{x} \\right) + f \\left( \\frac{1}{y} \\r... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 37888,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the intersection of the lines $9x-4y=30$ and $7x+y=11.$ Express your answer as an ordered pair $(x,y).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(2,-3)"
} | {
"split": "train",
"index": 37889,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $R$ be the set of all possible remainders when a number of the form $2^n$, $n$ a nonnegative integer, is divided by 1000. Let $S$ be the sum of the elements in $R$. Find the remainder when $S$ is divided by 1000.\n Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 37890,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the absolute value of the difference of single-digit integers $C$ and $D$ such that in base 8:\n$$ \\begin{array}{c@{}c@{\\;}c@{}c@{}c@{}c}\n & & & D & D & C_8 \\\\\n-& & & \\mathbf{6} & \\mathbf{3} & D_8 \\\\\n\\cline{2-6}\n& & C & \\mathbf{3} & \\mathbf{1} & \\mathbf{5_8... | math | {
"style": "rule",
"ground_truth": "5_8"
} | {
"split": "train",
"index": 37891,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a right triangle \\( ABC \\) with \\(\\angle A = 60^\\circ\\) and hypotenuse \\( AB = 2 + 2\\sqrt{3} \\), a line \\( p \\) is drawn through vertex \\( B \\) parallel to \\( AC \\). Points \\( D \\) and \\( E \\) are placed on line \\( p \\) such that \\( AB = BD \\) and \... | math | {
"style": "rule",
"ground_truth": "1 + \\sqrt{3} + \\sqrt{6}"
} | {
"split": "train",
"index": 37892,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$E$ is the midpoint of side $BC$ of parallelogram $ABCD$. Line $AE$ intersects the diagonal $BD$ at point $G$. If the area of triangle $\\triangle BEG$ is 1, find the area of parallelogram $ABCD$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 37893,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $-\\frac{\\pi}{2} < x < 0$ and $\\sin x + \\cos x = \\frac{1}{5}$.\n\n(I) Find the value of $\\sin x - \\cos x$.\n\n(II) Find the value of $\\frac{3\\sin^2{\\frac{x}{2}} - 2\\sin{\\frac{x}{2}}\\cos{\\frac{x}{2}} + \\cos^2{\\frac{x}{2}}}{\\tan x + \\cot x}$. Let's thi... | math | {
"style": "rule",
"ground_truth": "-\\frac{108}{125}"
} | {
"split": "train",
"index": 37894,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a round-robin tournament with 6 teams, each team plays one game against each other team, and each game results in one team winning and one team losing. At the end of the tournament, the teams are ranked by the number of games won. What is the maximum number of teams that cou... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 37895,
"solution": "1. **Calculate the total number of games**: In a round-robin tournament with 6 teams, each team plays against every other team exactly once. The total number of games played is given by the combination formula $\\binom{n}{2}$, where $n$ is the number of teams. Thus... |
deepscaler | [
{
"role": "user",
"content": "What is the difference between the largest and smallest numbers in the list $0.023,0.302,0.203,0.320,0.032$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.297"
} | {
"split": "train",
"index": 37896,
"solution": "We write the list in increasing order: $0.023,0.032,0.203,0.302,0.320$. The difference between the largest and smallest of these numbers is $0.320-0.023=0.297$."
} |
deepscaler | [
{
"role": "user",
"content": "\\frac{3 \\times 5}{9 \\times 11} \\times \\frac{7 \\times 9 \\times 11}{3 \\times 5 \\times 7}= Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 37897,
"solution": "1. **Identify the expression to simplify**: \n \\[\n \\frac{3 \\times 5}{9 \\times 11} \\times \\frac{7 \\times 9 \\times 11}{3 \\times 5 \\times 7}\n \\]\n\n2. **Rewrite the expression using the associative property**:\n The associative property allows us ... |
deepscaler | [
{
"role": "user",
"content": "Point \\( A \\) lies on the line \\( y = \\frac{5}{12} x - 7 \\), and point \\( B \\) lies on the parabola \\( y = x^2 \\). What is the minimum length of segment \\( AB \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4007}{624}"
} | {
"split": "train",
"index": 37898,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the ratio of the volume of cone $C$ to the volume of cone $D$? Cone $C$ has a radius of 15.6 and a height of 29.5. Cone $D$ has a radius of 29.5 and a height of 15.6. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{156}{295}"
} | {
"split": "train",
"index": 37899,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that an odd function \\( f(x) \\) satisfies the condition \\( f(x+3) = f(x) \\). When \\( x \\in [0,1] \\), \\( f(x) = 3^x - 1 \\). Find the value of \\( f\\left(\\log_1 36\\right) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1/3"
} | {
"split": "train",
"index": 37900,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of the expression $2 \\times 3 + 2 \\times 3$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 37901,
"solution": "Evaluating, $2 \\times 3 + 2 \\times 3 = 6 + 6 = 12$."
} |
deepscaler | [
{
"role": "user",
"content": "The sum of two natural numbers is $17402$. One of the two numbers is divisible by $10$. If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "14238"
} | {
"split": "train",
"index": 37902,
"solution": "1. **Identify the relationship between the two numbers**: Given that one number is divisible by $10$ and removing its units digit (which is $0$) gives the other number, we can denote the smaller number as $a$ and the larger number as $10a$.\n\n2. **Set up the equat... |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\prod_{n = 1}^{20} \\frac{n + 3}{n}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1771"
} | {
"split": "train",
"index": 37903,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum real number \\( M \\) such that for all real numbers \\( x \\) and \\( y \\) satisfying \\( x + y \\geqslant 0 \\), the following inequality holds:\n$$\n\\left(x^{2}+y^{2}\\right)^{3} \\geqslant M\\left(x^{3}+y^{3}\\right)(xy - x - y). Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 37904,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Use the Horner's method to calculate the value of the polynomial $f(x) = 2x^5 - 3x^2 + 4x^4 - 2x^3 + x$ when $x=2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "102"
} | {
"split": "train",
"index": 37905,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and it is given that $4\\cos C \\cdot \\sin^2 \\frac{C}{2} + \\cos 2C = 0$.\n\n(1) If $\\tan A = 2\\tan B$, find the value of $\\sin(A-B)$;\n\n(2) If $3ab = 25 - c^2$, find t... | math | {
"style": "rule",
"ground_truth": "\\frac{25\\sqrt{3}}{16}"
} | {
"split": "train",
"index": 37906,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a sequence of complex numbers $\\left\\{z_{n}\\right\\}$ defined as \"interesting\" if $\\left|z_{1}\\right|=1$ and for every positive integer $n$, the following holds:\n$$\n4 z_{n+1}^{2}+2 z_{n} z_{n+1}+z_{n}^{2}=0.\n$$\n\nFind the largest constant $C$ such that for a... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 37907,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A transgalactic ship encountered an astonishing meteor stream. Some meteors fly along a straight line at the same speed, equally spaced from each other. Another group of meteors flies in the exact same manner along another straight line, parallel to the first, but in the opposi... | math | {
"style": "rule",
"ground_truth": "4.6"
} | {
"split": "train",
"index": 37908,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What are the last three digits of \\(2003^N\\), where \\(N = 2002^{2001}\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "241"
} | {
"split": "train",
"index": 37909,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three circles, each of radius $3$, are drawn with centers at $(14, 92)$, $(17, 76)$, and $(19, 84)$. A line passing through $(17,76)$ is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circl... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 37910,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In right triangle $PQR$, $PQ=15$, $QR=8$, and angle $R$ is a right angle. A semicircle is inscribed in the triangle such that it touches $PQ$ and $QR$ at their midpoints and the hypotenuse $PR$. What is the radius of the semicircle?\nA) $\\frac{24}{5}$\n\nB) $\\frac{12}{5}$\n\n... | math | {
"style": "rule",
"ground_truth": "\\frac{24}{5}"
} | {
"split": "train",
"index": 37911,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ $c,$ $d$ be real numbers such that $a + b + c + d = 10$ and\n\\[ab + ac + ad + bc + bd + cd = 20.\\]\nFind the largest possible value of $d.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5 + \\sqrt{105}}{2}"
} | {
"split": "train",
"index": 37912,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer \\( n \\) such that for any set of \\( n \\) distinct integers \\( a_{1}, a_{2}, \\ldots, a_{n} \\), the product of all differences \\( a_{i} - a_{j} \\) for \\( i < j \\) is divisible by 1991. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "182"
} | {
"split": "train",
"index": 37913,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a two-lane highway where both lanes are single-directional, cars in both lanes travel at different constant speeds. The speed of cars in the left lane is 10 kilometers per hour higher than in the right lane. Cars follow a modified safety rule: the distance from the back of t... | math | {
"style": "rule",
"ground_truth": "338"
} | {
"split": "train",
"index": 37914,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Petya wants to color some cells of a $6 \\times 6$ square so that there are as many vertices as possible that belong to exactly three colored squares. What is the maximum number of such vertices he can achieve? Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 37915,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A dragon has 40 piles of gold coins, with the number of coins in any two piles differing. After the dragon plundered a neighboring city and brought back more gold, the number of coins in each pile increased by either 2, 3, or 4 times. What is the minimum number of different pil... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 37916,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A person flips $2010$ coins at a time. He gains one penny every time he flips a prime number of heads, but must stop once he flips a non-prime number. If his expected amount of money gained in dollars is $\\frac{a}{b}$ , where $a$ and $b$ are relatively prime, compute $... | math | {
"style": "rule",
"ground_truth": "2017"
} | {
"split": "train",
"index": 37917,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( N \\) be the positive integers. The function \\( f : N \\to N \\) satisfies \\( f(1) = 5 \\), \\( f(f(n)) = 4n + 9 \\), and \\( f(2n) = 2n+1 + 3 \\) for all \\( n \\). Find \\( f(1789) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3581"
} | {
"split": "train",
"index": 37918,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the largest three-digit multiple of 4 and the smallest four-digit multiple of 3? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1998"
} | {
"split": "train",
"index": 37919,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given triangle $ABC$ with vertices $A = (3,0)$, $B = (0,3)$, and $C$ lying on the line $x + 2y = 8$, find the area of triangle $ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"split": "train",
"index": 37920,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The terms of the sequence $(a_i)$ defined by $a_{n + 2} = \\frac {a_n + 2009} {1 + a_{n + 1}}$ for $n \\ge 1$ are positive integers. Find the minimum possible value of $a_1 + a_2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 37921,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n \\geq 5$ be a given integer. Determine the greatest integer $k$ for which there exists a polygon with $n$ vertices (convex or not, with non-selfintersecting boundary) having $k$ internal right angles.\n\n[i] Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "g(n)=\\lceil\\frac{2n+1}{3}\\rceil"
} | {
"split": "train",
"index": 37922,
"solution": "\nLet \\( n \\geq 5 \\) be a given integer. We are tasked with determining the greatest integer \\( k \\) for which there exists a polygon with \\( n \\) vertices (which can be either convex or non-convex, with a non-self-intersecting boundary) having \\( k \\) int... |
deepscaler | [
{
"role": "user",
"content": "How many $6$ -digit positive integers have their digits in nondecreasing order from left to right? Note that $0$ cannot be a leading digit. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3003"
} | {
"split": "train",
"index": 37923,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a natural number $n \\ge 3$ , we draw $n - 3$ internal diagonals in a non self-intersecting, but not necessarily convex, n-gon, cutting the $n$ -gon into $n - 2$ triangles. It is known that the value (in degrees) of any angle in any of these triangles is a natural num... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 37924,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the three digits 2, 4 and 7, how many different positive two-digit integers can be formed using these digits if a digit may not be repeated in an integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 37925,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\angle ABC = 24^\\circ$ and $\\angle ABD = 20^\\circ$. What is the smallest possible degree measure for $\\angle CBD$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 37926,
"solution": "1. **Identify the Relationship Between Angles**: Given $\\angle ABC = 24^\\circ$ and $\\angle ABD = 20^\\circ$, we know that these two angles share the common ray $AB$. \n\n2. **Expression for $\\angle CBD$**: Since $\\angle ABC$ is the full angle formed at point $... |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle ABC$ be equilateral with integer side length. Point $X$ lies on $\\overline{BC}$ strictly between $B$ and $C$ such that $BX<CX$ . Let $C'$ denote the reflection of $C$ over the midpoint of $\\overline{AX}$ . If $BC'=30$ , find the sum of all pos... | math | {
"style": "rule",
"ground_truth": "130"
} | {
"split": "train",
"index": 37927,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Insert a square into an isosceles triangle with a lateral side of 10 and a base of 12. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4.8"
} | {
"split": "train",
"index": 37928,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $AB + AC$ in triangle $ABC$ given that $D$ is the midpoint of $BC$, $E$ is the midpoint of $DC$, and $BD = DE = EA = AD$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1+\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 37929,
"solution": "$DBC$ is a right triangle with hypotenuse $DC$. Since $DE=EC$, $E$ is the midpoint of this right triangle's hypotenuse, and it follows that $E$ is the circumcenter of the triangle. It follows that $BE=DE=CE$, as these are all radii of the same circle. A similar arg... |
deepscaler | [
{
"role": "user",
"content": "Let \\[f(x) =\n\\begin{cases}\n3x + 5 &\\text{if }x<-3, \\\\\n7-4x&\\text{if }x\\ge -3.\n\\end{cases}\n\\]Find $f(5)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-13"
} | {
"split": "train",
"index": 37930,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the points $A(0,12), B(10,9), C(8,0),$ and $D(-4,7).$ There is a unique square $S$ such that each of the four points is on a different side of $S.$ Let $K$ be the area of $S.$ Find the remainder when $10K$ is divided by $1000$. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "936"
} | {
"split": "train",
"index": 37931,
"solution": "Let $(a,b)$ denote a normal vector of the side containing $A$. Note that $\\overline{AC}, \\overline{BD}$ intersect and hence must be opposite vertices of the square. The lines containing the sides of the square have the form $ax+by=12b$, $ax+by=8a$, $bx-ay=10b-9a$... |
deepscaler | [
{
"role": "user",
"content": "Simplify: $\\sqrt{50} + \\sqrt{18}$ . Express your answer in simplest radical form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8\\sqrt{2}"
} | {
"split": "train",
"index": 37932,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $α∈(\\frac{\\pi}{2},π)$, $\\sin α =\\frac{\\sqrt{5}}{5}$.\n\n(Ⅰ) Find the value of $\\tan\\left( \\frac{\\pi}{4}+2α \\right)$;\n\n(Ⅱ) Find the value of $\\cos\\left( \\frac{5\\pi}{6}-2α \\right)$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "-\\frac{3\\sqrt{3}+4}{10}"
} | {
"split": "train",
"index": 37933,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the sum of all positive integers $n$ for which $9 \\sqrt{n}+4 \\sqrt{n+2}-3 \\sqrt{n+16}$ is an integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 37934,
"solution": "For the expression to be an integer at least one of $n$ and $n+2$ must be a perfect square. We also note that at most one of $n$ and $n+2$ can be a square, so exactly one of them is a square. Case 1: $n$ is a perfect square. By our previous observation, it must be ... |
deepscaler | [
{
"role": "user",
"content": "From the set $\\{1, 2, 3, 4, \\ldots, 20\\}$, select four different numbers $a, b, c, d$ such that $a+c=b+d$. If the order of $a, b, c, d$ does not matter, calculate the total number of ways to select these numbers. Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "525"
} | {
"split": "train",
"index": 37935,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all positive real numbers $x$ that satisfy\n\\[x \\sqrt{12 - x} + \\sqrt{12x - x^3} \\ge 12.\\]Enter all solutions, separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 37936,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$, $b$, $c$ be the three sides of a triangle, and let $\\alpha$, $\\beta$, $\\gamma$, be the angles opposite them. If $a^2+b^2=1989c^2$, find\n$\\frac{\\cot \\gamma}{\\cot \\alpha+\\cot \\beta}$ Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "994"
} | {
"split": "train",
"index": 37937,
"solution": "We start as in solution 1, though we'll write $A$ instead of $K$ for the area. Now we evaluate the numerator:\n\\[\\cot{\\gamma}=\\frac{\\cos{\\gamma}}{\\sin{\\gamma}}\\]\n From the Law of Cosines and the sine area formula,\n\\begin{align*}\\cos{\\gamma}&=\\frac{19... |
deepscaler | [
{
"role": "user",
"content": "Given the parametric equation of line $l$ as $$\\begin{cases} x= \\sqrt {3}+t \\\\ y=7+ \\sqrt {3}t\\end{cases}$$ ($t$ is the parameter), a coordinate system is established with the origin as the pole and the positive half of the $x$-axis as the polar axis. The polar equation o... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{21}}{3}"
} | {
"split": "train",
"index": 37938,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points \\( A, B, C \\) are situated sequentially, with the distance \\( AB \\) being 3 km and the distance \\( BC \\) being 4 km. A cyclist departed from point \\( A \\) heading towards point \\( C \\). Simultaneously, a pedestrian departed from point \\( B \\) heading towards ... | math | {
"style": "rule",
"ground_truth": "2.1"
} | {
"split": "train",
"index": 37939,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the sequence $a_{1}, a_{2}, \\cdots$ be defined recursively as follows: $a_{n}=11a_{n-1}-n$ . If all terms of the sequence are positive, the smallest possible value of $a_{1}$ can be written as $\\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integ... | math | {
"style": "rule",
"ground_truth": "121"
} | {
"split": "train",
"index": 37940,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the $xy$-plane, consider a T-shaped region bounded by horizontal and vertical segments with vertices at $(0,0), (0,4), (4,4), (4,2), (7,2), (7,0)$. What is the slope of the line through the origin that divides the area of this region exactly in half?\nA) $\\frac{1}{4}$\nB) $... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 37941,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pyramid has a square base $ABCD$ and vertex $E$. The area of square $ABCD$ is $196$, and the areas of $\\triangle ABE$ and $\\triangle CDE$ are $105$ and $91$, respectively. What is the volume of the pyramid? Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "784"
} | {
"split": "train",
"index": 37942,
"solution": "1. **Identify the side length of the square base**: Given the area of square $ABCD$ is $196$, we find the side length by taking the square root:\n \\[\n s = \\sqrt{196} = 14\n \\]\n\n2. **Calculate the heights of triangles $ABE$ and $CDE$**: The area of a tri... |
deepscaler | [
{
"role": "user",
"content": "A woman buys a property for $150,000 with a goal to achieve a $7\\%$ annual return on her investment. She sets aside $15\\%$ of each month's rent for maintenance costs, and pays property taxes at $0.75\\%$ of the property's value each year. Calculate the monthly rent she needs ... | math | {
"style": "rule",
"ground_truth": "1139.71"
} | {
"split": "train",
"index": 37943,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the function f(x) defined on the set of real numbers ℝ satisfies f(x+1) = 1/2 + √(f(x) - f^2(x)), find the maximum value of f(0) + f(2017). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1+\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 37944,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of all points with coordinates $(x,y,z)$, where $x$, $y$, and $z$ are each chosen from the set $\\{0,1,2\\}$. How many equilateral triangles have all their vertices in $S$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 37945,
"solution": "\n#### Step 1: Understanding the Set $S$\nThe set $S$ consists of all points $(x,y,z)$ where $x, y, z \\in \\{0,1,2\\}$. This forms a $3 \\times 3 \\times 3$ grid of points in three-dimensional space.\n\n#### Step 2: Possible Distances Between Points\nUsing the thr... |
deepscaler | [
{
"role": "user",
"content": "Compute $\\tan 0^\\circ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 37946,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a+b = 6$ and $a - b = 2$, what is the value of $a^2 - b^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 37947,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In chess tournaments, each victory is worth 1 point, each draw is worth 0.5 points, and each loss is worth zero points. In the \"Magistral Championship\" of chess, only International Masters (IMs) and Grand Masters (GMs) participated. The number of GMs was ten times the number ... | math | {
"style": "rule",
"ground_truth": "55"
} | {
"split": "train",
"index": 37948,
"solution": ""
} |
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