data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Given the rectangular coordinate system xOy, establish a polar coordinate system with O as the pole and the non-negative semi-axis of the x-axis as the polar axis. The line l passes through point P(-1, 2) with an inclination angle of $\\frac{2π}{3}$, and the polar coordinate eq... | math | {
"style": "rule",
"ground_truth": "6 + 2\\sqrt{3}"
} | {
"split": "train",
"index": 29449,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xoy$, the parametric equation of curve $C_1$ is \n$$\n\\begin{cases} \nx=2\\sqrt{2}-\\frac{\\sqrt{2}}{2}t \\\\\ny=\\sqrt{2}+\\frac{\\sqrt{2}}{2}t\n\\end{cases}\n(t \\text{ is the parameter}).\n$$\nIn the polar coordinate system with the origi... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{7}"
} | {
"split": "train",
"index": 29450,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( n \\) numbers \\( a_{1}, a_{2}, \\cdots, a_{n} \\), their root mean square is defined as \\(\\left(\\frac{a_{1}^{2}+a_{2}^{2}+\\cdots+a_{n}^{2}}{n}\\right)^{\\frac{1}{2}} \\). Let \\( M \\) be the set of values of \\( n \\) (for \\( n > 1 \\)) such that the root mean ... | math | {
"style": "rule",
"ground_truth": "337"
} | {
"split": "train",
"index": 29451,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular octahedron is formed by joining the centers of adjoining faces of a cube. The ratio of the volume of the octahedron to the volume of the cube is\n$\\mathrm{(A) \\frac{\\sqrt{3}}{12} } \\qquad \\mathrm{(B) \\frac{\\sqrt{6}}{16} } \\qquad \\mathrm{(C) \\frac{1}{6} } \\q... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 29452,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The solution of the equation $\\dfrac{1+2^{x}}{1+x^{-x}}= \\dfrac{1}{4}$ is $x=$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 29453,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a linear function \\( f(x) \\), it is known that the distance between the intersection points of the graphs \\( y=x^{2}+2 \\) and \\( y=f(x) \\) is \\(\\sqrt{10}\\), and the distance between the intersection points of the graphs \\( y=x^{2}-1 \\) and \\( y=f(x)+1 \\) is \... | math | {
"style": "rule",
"ground_truth": "\\sqrt{34}"
} | {
"split": "train",
"index": 29454,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that in $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $b\\sin A+a\\cos B=0$.\n(1) Find the measure of angle $B$;\n(2) If $b=2$, find the maximum area of $\\triangle ABC$. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}-1"
} | {
"split": "train",
"index": 29455,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all $x$ that satisfy the equation $\\frac{-9x}{x^2-1} = \\frac{2x}{x+1} - \\frac{6}{x-1}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 29456,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equation of the line joining the complex numbers $-1 + 2i$ and $2 + 3i$ can be expressed in the form\n\\[az + b \\overline{z} = d\\]for some complex numbers $a$, $b$, and real number $d$. Find the product $ab$. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 29457,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 16 different cards, including 4 red, 4 yellow, 4 blue, and 4 green cards. If 3 cards are drawn at random, the requirement is that these 3 cards cannot all be of the same color, and at most 1 red card is allowed. The number of different ways to draw the cards is \\_\\_... | math | {
"style": "rule",
"ground_truth": "472"
} | {
"split": "train",
"index": 29458,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express the quotient $1023_4 \\div 11_4$ in base 4. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "33_4"
} | {
"split": "train",
"index": 29459,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The bases \\(AB\\) and \\(CD\\) of the trapezoid \\(ABCD\\) are 367 and 6 respectively, and its diagonals are mutually perpendicular. Find the scalar product of the vectors \\(\\overrightarrow{AD}\\) and \\(\\overrightarrow{BC}\\). Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "2202"
} | {
"split": "train",
"index": 29460,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "As shown in the figure, triangle $ABC$ is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles are as indicated. Find the area of triangle $ABC$. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "315"
} | {
"split": "train",
"index": 29461,
"solution": "This problem can be done using mass points. Assign B a weight of 1 and realize that many of the triangles have the same altitude. After continuously using the formulas that state (The sum of the two weights) = (The middle weight), and (The weight $\\times$ side) = ... |
deepscaler | [
{
"role": "user",
"content": "In trapezoid $ABCD$ the lengths of the bases $AB$ and $CD$ are 8 and 17 respectively. The legs of the trapezoid are extended beyond $A$ and $B$ to meet at point $E$. What is the ratio of the area of triangle $EAB$ to the area of trapezoid $ABCD$? Express your answer as a common... | math | {
"style": "rule",
"ground_truth": "\\frac{64}{225}"
} | {
"split": "train",
"index": 29462,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $52103_{8} - 1452_{9}$ in base 10. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20471"
} | {
"split": "train",
"index": 29463,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Originally, every square of $8 \\times 8$ chessboard contains a rook. One by one, rooks which attack an odd number of others are removed. Find the maximal number of rooks that can be removed. (A rook attacks another rook if they are on the same row or column and there are no ot... | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 29464,
"solution": "\nGiven an \\(8 \\times 8\\) chessboard where each square initially contains a rook, we need to determine the maximal number of rooks that can be removed such that each removed rook initially attacked an odd number of other rooks. A rook attacks another rook if the... |
deepscaler | [
{
"role": "user",
"content": "If $\\alpha \\in (0, \\frac{\\pi}{2})$, and $\\tan 2\\alpha = \\frac{\\cos \\alpha}{2-\\sin \\alpha}$, then find the value of $\\tan \\alpha$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{15}}{15}"
} | {
"split": "train",
"index": 29465,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N$ be the largest positive integer with the following property: reading from left to right, each pair of consecutive digits of $N$ forms a perfect square. What are the leftmost three digits of $N$? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "816"
} | {
"split": "train",
"index": 29466,
"solution": "The two-digit perfect squares are $16, 25, 36, 49, 64, 81$. We try making a sequence starting with each one:\n$16 - 64 - 49$. This terminates since none of them end in a $9$, giving us $1649$.\n$25$.\n$36 - 64 - 49$, $3649$.\n$49$.\n$64 - 49$, $649$.\n$81 - 16 - 64... |
deepscaler | [
{
"role": "user",
"content": "Let $a,b,c$ be positive real numbers such that $a+b+c = 3$. Find the minimum value of the expression \\[A=\\dfrac{2-a^3}a+\\dfrac{2-b^3}b+\\dfrac{2-c^3}c.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 29467,
"solution": "\nTo find the minimum value of the expression \n\\[ \nA = \\frac{2-a^3}{a} + \\frac{2-b^3}{b} + \\frac{2-c^3}{c}, \n\\]\ngiven that \\( a, b, c \\) are positive real numbers and \\( a + b + c = 3 \\), we proceed as follows:\n\nFirst, we rewrite the expression:\n\\[... |
deepscaler | [
{
"role": "user",
"content": "Choose $3$ different numbers from the $5$ numbers $0$, $1$, $2$, $3$, $4$ to form a three-digit even number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 29468,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate \\( \\left[6 \\frac{3}{5}-\\left(8.5-\\frac{1}{3}\\right) \\div 3.5\\right] \\times\\left(2 \\frac{5}{18}+\\frac{11}{12}\\right) = \\) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{368}{27}"
} | {
"split": "train",
"index": 29469,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Natural numbers \\( a, b, c \\) are chosen such that \\( a < b < c \\). It is also known that the system of equations \\( 2x + y = 2025 \\) and \\( y = |x - a| + |x - b| + |x - c| \\) has exactly one solution. Find the minimum possible value of \\( c \\). Let's think step by st... | math | {
"style": "rule",
"ground_truth": "1013"
} | {
"split": "train",
"index": 29470,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the letters a, b, c, d, e arranged in a row, find the number of arrangements where both a and b are not adjacent to c. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 29471,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the radius of circle \\( \\odot O \\) is 1, the quadrilateral \\( ABCD \\) is an inscribed square, \\( EF \\) is a diameter of \\( \\odot O \\), and \\( M \\) is a point moving along the boundary of the square \\( ABCD \\). Find the minimum value of \\(\\overrightarr... | math | {
"style": "rule",
"ground_truth": "-1/2"
} | {
"split": "train",
"index": 29472,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle ABC$ have side lengths $AB=30$, $BC=32$, and $AC=34$. Point $X$ lies in the interior of $\\overline{BC}$, and points $I_1$ and $I_2$ are the incenters of $\\triangle ABX$ and $\\triangle ACX$, respectively. Find the minimum possible area of $\\triangle AI_1I_2$ ... | math | {
"style": "rule",
"ground_truth": "126"
} | {
"split": "train",
"index": 29473,
"solution": "First note that \\[\\angle I_1AI_2 = \\angle I_1AX + \\angle XAI_2 = \\frac{\\angle BAX}2 + \\frac{\\angle CAX}2 = \\frac{\\angle A}2\\] is a constant not depending on $X$, so by $[AI_1I_2] = \\tfrac12(AI_1)(AI_2)\\sin\\angle I_1AI_2$ it suffices to minimize $(AI_1... |
deepscaler | [
{
"role": "user",
"content": "Two railway tracks intersect at a right angle. Two trains are simultaneously speeding towards the intersection point from different tracks: one from a station located 40 km from the intersection point, and the other from a station located 50 km away. The first train travels at ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 29474,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The factory's planned output value for this year is $a$ million yuan, which is a 10% increase from last year. If the actual output value this year can exceed the plan by 1%, calculate the increase in the actual output value compared to last year. Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "11.1\\%"
} | {
"split": "train",
"index": 29475,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Complex numbers $a,$ $b,$ and $c$ are zeros of a polynomial $P(z) = z^3 + qz + r,$ and $|a|^2 + |b|^2 + |c|^2 = 250.$ The points corresponding to $a,$ $b,$ and $c$ in the complex plane are the vertices of a right triangle with hypotenuse $h.$ Find $h^2.$ Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "375"
} | {
"split": "train",
"index": 29476,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the number of ordered pairs of integers $(x, y)$ such that $x^{2}+y^{2}<2019$ and $$x^{2}+\\min (x, y)=y^{2}+\\max (x, y)$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "127"
} | {
"split": "train",
"index": 29477,
"solution": "We have $$x^{2}-y^{2}=\\max (x, y)-\\min (x, y)=|x-y|$$ Now if $x \\neq y$, we can divide by $x-y$ to obtain $x+y= \\pm 1$. Thus $x=y$ or $x+y= \\pm 1$. If $x=y$, we see that $2019>x^{2}+y^{2}=2 x^{2}$, so we see that $-31 \\leq x \\leq 31$. There are 63 ordered pa... |
deepscaler | [
{
"role": "user",
"content": "Tessa picks three real numbers $x, y, z$ and computes the values of the eight expressions of the form $\\pm x \\pm y \\pm z$. She notices that the eight values are all distinct, so she writes the expressions down in increasing order. How many possible orders are there? Let's th... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 29478,
"solution": "There are $2^{3}=8$ ways to choose the sign for each of $x, y$, and $z$. Furthermore, we can order $|x|,|y|$, and $|z|$ in $3!=6$ different ways. Now assume without loss of generality that $0<x<y<z$. Then there are only two possible orders depending on the sign of ... |
deepscaler | [
{
"role": "user",
"content": "Ken is the best sugar cube retailer in the nation. Trevor, who loves sugar, is coming over to make an order. Ken knows Trevor cannot afford more than 127 sugar cubes, but might ask for any number of cubes less than or equal to that. Ken prepares seven cups of cubes, with which ... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 29479,
"solution": "The only way to fill seven cups to satisfy the above condition is to use a binary scheme, so the cups must contain $1,2,4,8,16,32$, and 64 cubes of sugar."
} |
deepscaler | [
{
"role": "user",
"content": "Given real numbers \\(a, b, c\\), the polynomial \n$$\ng(x) = x^{3} + a x^{2} + x + 10\n$$ \nhas three distinct roots, and these three roots are also roots of the polynomial\n$$\nf(x) = x^{4} + x^{3} + b x^{2} + 100 x + c.\n$$\n\nFind the value of \\(f(1)\\). Let's think step b... | math | {
"style": "rule",
"ground_truth": "-7007"
} | {
"split": "train",
"index": 29480,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagonals \\(AC\\) and \\(BD\\) of the symmetric trapezoid \\(ABCD\\) intersect at point \\(O\\). The area of triangle \\(AOB\\) is \\(52 \\, \\text{m}^2\\) and the area of triangle \\(COD\\) is \\(117 \\, \\text{m}^2\\). Calculate the area of the trapezoid. Let's think ste... | math | {
"style": "rule",
"ground_truth": "325"
} | {
"split": "train",
"index": 29481,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nPoints are marked on a circle, dividing it into 2012 equal arcs. From these, $k$ points are chosen to construct a convex $k$-gon with vertices at the chosen points. What is the maximum possible value of $k$ such that this polygon has no parallel sides? Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "1509"
} | {
"split": "train",
"index": 29482,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cards numbered 1 and 2 must be placed into the same envelope, and three cards are left to be placed into two remaining envelopes, find the total number of different methods. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 29483,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is 57, how many tiles cover the floor. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "841"
} | {
"split": "train",
"index": 29484,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sandy's daughter has a playhouse in the back yard. She plans to cover the one shaded exterior wall and the two rectangular faces of the roof, also shaded, with a special siding to resist the elements. The siding is sold only in 8-foot by 12-foot sections that cost $\\$27.30$ ea... | math | {
"style": "rule",
"ground_truth": "\\$ 54.60"
} | {
"split": "train",
"index": 29485,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the regular hexagon \\(ABCDEF\\), two of the diagonals, \\(FC\\) and \\(BD\\), intersect at \\(G\\). The ratio of the area of quadrilateral \\(FEDG\\) to \\(\\triangle BCG\\) is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5: 1"
} | {
"split": "train",
"index": 29486,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of pairs of positive integers $x,y$ such that $x\\le y$ , $\\gcd (x,y)=5!$ and $\\text{lcm}(x,y)=50!$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16384"
} | {
"split": "train",
"index": 29487,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $α \\in \\left( \\frac{π}{2}, π \\right)$, and $\\sin α = \\frac{1}{3}$.\n\n$(1)$ Find the value of $\\sin 2α$; \n$(2)$ If $\\sin (α+β) = -\\frac{3}{5}$, and $β \\in (0, \\frac{π}{2})$, find the value of $\\sin β$. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "\\frac{6\\sqrt{2}+4}{15}"
} | {
"split": "train",
"index": 29488,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Tony works $2$ hours a day and is paid $\\$0.50$ per hour for each full year of his age. During a six month period Tony worked $50$ days and earned $\\$630$. How old was Tony at the end of the six month period? Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 29489,
"solution": "1. **Understanding the payment structure**: Tony earns $0.50$ per hour for each full year of his age. Since he works $2$ hours a day, his daily earnings are $2 \\times 0.50 \\times \\text{age} = \\text{age}$ dollars per day.\n\n2. **Setting up the equation**: Tony ... |
deepscaler | [
{
"role": "user",
"content": "Point $C$ is on the segment $AB$ which has endpoints $A(-1, 0)$ and $B(3, 8)$. Point $C$ is three times as far from point $A$ as it is from point $B$. What are the coordinates of point $C$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(2,6)"
} | {
"split": "train",
"index": 29490,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The minimum value of \\\\(f(x)=\\sin x+\\cos x-\\sin x\\cos x\\\\) is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{1}{2}- \\sqrt{2}"
} | {
"split": "train",
"index": 29491,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the expression \\(\\frac{a}{b}+\\frac{c}{d}+\\frac{e}{f}\\), where each letter is replaced by a different digit from \\(1, 2, 3, 4, 5,\\) and \\(6\\), determine the largest possible value of this expression. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "9\\frac{5}{6}"
} | {
"split": "train",
"index": 29492,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $(2x^2 + 7x - 3) - (x^2 + 5x - 12)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x^2+2x+9"
} | {
"split": "train",
"index": 29493,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the random variable $\\xi$ follows a normal distribution $N(2, \\sigma^2)$, and $P(\\xi < 3) = 0.6$, find $P(1 < \\xi < 2)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.1"
} | {
"split": "train",
"index": 29494,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\cos x \\cdot \\sin \\left( x + \\frac{\\pi}{3} \\right) - \\sqrt{3}\\cos^2 x + \\frac{\\sqrt{3}}{4}, x \\in \\mathbb{R}$,\n\n$(1)$ Find the smallest positive period of $f(x)$;\n\n$(2)$ Find the maximum and minimum values of $f(x)$ on the closed inte... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 29495,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Teacher Li and three students (Xiao Ma, Xiao Lu, and Xiao Zhou) depart from school one after another and walk the same route to a cinema. The walking speed of the three students is equal, and Teacher Li's walking speed is 1.5 times that of the students. Currently, Teacher Li is... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 29496,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S_{0}=0$ and let $S_{k}$ equal $a_{1}+2 a_{2}+\\ldots+k a_{k}$ for $k \\geq 1$. Define $a_{i}$ to be 1 if $S_{i-1}<i$ and -1 if $S_{i-1} \\geq i$. What is the largest $k \\leq 2010$ such that $S_{k}=0$? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "1092"
} | {
"split": "train",
"index": 29497,
"solution": "Suppose that $S_{N}=0$ for some $N \\geq 0$. Then $a_{N+1}=1$ because $N+1 \\geq S_{N}$. The following table lists the values of $a_{k}$ and $S_{k}$ for a few $k \\geq N$: $k$ & $a_{k}$ & $S_{k}$ \\hline$N$ & & 0 \\$N+1$ & 1 & $N+1$ \\$N+2$ & 1 & $2 N+3$ \\$N+3$ & ... |
deepscaler | [
{
"role": "user",
"content": "A smaller square was cut out from a larger square in such a way that one side of the smaller square lies on a side of the original square. The perimeter of the resulting octagon is $40\\%$ greater than the perimeter of the original square. By what percentage is the area of the ... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 29498,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the table showing the weekly reading times of $30$ students, with $7$ students reading $6$ hours, $8$ students reading $7$ hours, $5$ students reading $8$ hours, and $10$ students reading $9$ hours, find the median of the weekly reading times for these $30$ students. Let'... | math | {
"style": "rule",
"ground_truth": "7.5"
} | {
"split": "train",
"index": 29499,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jennifer is hiking in the mountains. She walks northward for 3 miles, then turns 45 degrees eastward and walks 5 miles. How far is she from her starting point? 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": "\\sqrt{34 + 15\\sqrt{2}}"
} | {
"split": "train",
"index": 29500,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a set of $36$ square blocks arranged into a $6 \\times 6$ square, how many different combinations of $4$ blocks can be selected from that set so that no two are in the same row or column? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5400"
} | {
"split": "train",
"index": 29501,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a cube \\( ABCD A_1 B_1 C_1 D_1 \\) with an edge length of 1. A sphere passes through vertices \\( A \\) and \\( C \\) and the midpoints \\( F \\) and \\( E \\) of edges \\( B_1 C_1 \\) and \\( C_1 D_1 \\) respectively. Find the radius \\( R \\) of this sphere. Let's thin... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{41}}{8}"
} | {
"split": "train",
"index": 29502,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can a President, a Vice-President, and a Secretary be chosen from a group of 6 people, given the following constraints:\n- The President must be one of the first three members of the group.\n- The Vice-President must be one of the last four members of the group... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 29503,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $P$ is $\\sqrt3$ units away from plane $A$ . Let $Q$ be a region of $A$ such that every line through $P$ that intersects $A$ in $Q$ intersects $A$ at an angle between $30^o$ and $60^o$ . What is the largest possible area of $Q$ ? Let's think step by ... | math | {
"style": "rule",
"ground_truth": "8\\pi"
} | {
"split": "train",
"index": 29504,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many three-digit whole numbers have at least one 8 or at least one 9 as digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "452"
} | {
"split": "train",
"index": 29505,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $z$ be a complex number. In the complex plane, the distance from $z$ to 1 is 2 , and the distance from $z^{2}$ to 1 is 6 . What is the real part of $z$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{4}"
} | {
"split": "train",
"index": 29506,
"solution": "Note that we must have $|z-1|=2$ and \\left|z^{2}-1\\right|=6$, so $|z+1|=\\frac{\\left|z^{2}-1\\right|}{|z-1|}=3$. Thus, the distance from $z$ to 1 in the complex plane is 2 and the distance from $z$ to -1 in the complex plane is 3 . Thus, $z, 1,-1$ form a triangl... |
deepscaler | [
{
"role": "user",
"content": "A plane parallel to the base of a cone divides the height of the cone into two equal segments. What is the ratio of the lateral surface areas of the two parts of the cone? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 29507,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The triangle $ABC$ is isosceles with $AB=AC$, and $\\angle{BAC}<60^{\\circ}$. The points $D$ and $E$ are chosen on the side $AC$ such that, $EB=ED$, and $\\angle{ABD}\\equiv\\angle{CBE}$. Denote by $O$ the intersection point between the internal bisectors of the angles $\\angle... | math | {
"style": "rule",
"ground_truth": "120^\\circ"
} | {
"split": "train",
"index": 29508,
"solution": "\nGiven triangle \\( \\triangle ABC \\), where \\( AB = AC \\) and \\( \\angle BAC < 60^\\circ \\), points \\( D \\) and \\( E \\) are chosen on side \\( AC \\) such that \\( EB = ED \\) and \\( \\angle ABD \\equiv \\angle CBE \\). We are tasked with finding \\(\\a... |
deepscaler | [
{
"role": "user",
"content": "What describes the set of values of $a$ for which the curves $x^2+y^2=a^2$ and $y=x^2-a$ in the real $xy$-plane intersect at exactly 3 points? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "a>\\frac{1}{2}"
} | {
"split": "train",
"index": 29509,
"solution": "To solve this problem, we need to analyze the intersection points of the circle $x^2 + y^2 = a^2$ and the parabola $y = x^2 - a$. \n\n1. **Substitute the expression for $y$ from the parabola into the circle equation:**\n \\[\n x^2 + (x^2 - a)^2 = a^2\n \\]\n ... |
deepscaler | [
{
"role": "user",
"content": "The graph of $y=ax^2+bx+c$ is given below, where $a$, $b$, and $c$ are integers. Find $a-b+c$.\n\n[asy]\nsize(150);\n\nLabel f;\n\nf.p=fontsize(4);\n\nxaxis(-3,3,Ticks(f, 1.0));\n\nyaxis(-4,4,Ticks(f, 1.0));\n\nreal f(real x)\n\n{\n\nreturn x^2+2x-1;\n\n}\n\ndraw(graph(f,-2.7,... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 29510,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $2 x^{2}=9 x-4$ and $x \neq 4$, what is the value of $2 x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 29511,
"solution": "Since $2 x^{2}=9 x-4$, then $2 x^{2}-9 x+4=0$. Factoring, we obtain $(2 x-1)(x-4)=0$. Thus, $2 x=1$ or $x=4$. Since $x \neq 4$, then $2 x=1$."
} |
deepscaler | [
{
"role": "user",
"content": "Given that three cultural courses (Chinese, Mathematics, and Foreign Language) and three other arts courses are randomly scheduled in six periods, find the probability that no two adjacent cultural courses are separated by more than one arts course. Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "\\dfrac{3}{5}"
} | {
"split": "train",
"index": 29512,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The adult human body has 206 bones. Each foot has 26 bones. Approximately what fraction of the number of bones in the human body is found in one foot? 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": 29513,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two machine tools, A and B, produce the same product. The products are divided into first-class and second-class according to quality. In order to compare the quality of the products produced by the two machine tools, each machine tool produced 200 products. The quality of the ... | math | {
"style": "rule",
"ground_truth": "99\\%"
} | {
"split": "train",
"index": 29514,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the geometric sequence ${a_n}$, $a_3$ and $a_{15}$ are the roots of the equation $x^2 + 6x + 2 = 0$, calculate the value of $$\\frac{a_{2}a_{16}}{a_{9}}.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 29515,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In right triangle $\\triangle ABC$ with hypotenuse $\\overline{AB}$, $AC = 15$, $BC = 20$, and $\\overline{CD}$ is the altitude to $\\overline{AB}$. Let $\\omega$ be the circle having $\\overline{CD}$ as a diameter. Let $I$ be a point outside $\\triangle ABC$ such that $\\overl... | math | {
"style": "rule",
"ground_truth": "97"
} | {
"split": "train",
"index": 29516,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ a $ , $ b $ , $ c $ , $ d $ , $ (a + b + c + 18 + d) $ , $ (a + b + c + 18 - d) $ , $ (b + c) $ , and $ (c + d) $ be distinct prime numbers such that $ a + b + c = 2010 $ , $ a $ , $ b $ , $ c $ , $ d \\neq 3 $ , and $ d \\le 50 $ . Find the maximu... | math | {
"style": "rule",
"ground_truth": "2067"
} | {
"split": "train",
"index": 29517,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles A, B, and C are denoted as $a$, $b$, and $c$ respectively. Given that $a= \\sqrt{2}$, $b=2$, and $\\sin B - \\cos B = \\sqrt{2}$, find the measure of angle $A$. Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{6}"
} | {
"split": "train",
"index": 29518,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\frac{8xy^2}{6x^2y}$ with $x=2$ and $y=3.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 29519,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $N$ permutations $(a_{1}, a_{2}, ... , a_{30})$ of $1, 2, \\ldots, 30$ such that for $m \\in \\left\\{{2, 3, 5}\\right\\}$, $m$ divides $a_{n+m} - a_{n}$ for all integers $n$ with $1 \\leq n < n+m \\leq 30$. Find the remainder when $N$ is divided by $1000$. Let's thin... | math | {
"style": "rule",
"ground_truth": "440"
} | {
"split": "train",
"index": 29520,
"solution": "Note that $30=2\\cdot 3\\cdot 5$. Since $\\gcd(2, 3, 5)=1$, by CRT, for each value $k=0\\ldots 29$ modulo $30$ there exists a unique ordered triple of values $(a, b, c)$ such that $k\\equiv a\\pmod{2}$, $k\\equiv b\\pmod{3}$, and $k\\equiv c\\pmod{5}$. Therefore, w... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to the three internal angles are $a$, $b$, and $c$, respectively. Given that $\\cos A= \\frac{ \\sqrt {10}}{10}$ and $a\\sin A+b\\sin B-c\\sin C= \\frac{ 2 \\sqrt {5}}{5}a\\sin B$.\n1. Find the value of $B$;\n2. If $b=10$, find the area $... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 29521,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Real numbers \\(a, b, c\\) and a positive number \\(\\lambda\\) such that \\(f(x) = x^{3} + ax^{2} + bx + c\\) has three real roots \\(x_{1}, x_{2}, x_{3}\\), and satisfy:\n(1) \\(x_{2} - x_{1} = \\lambda\\);\n(2) \\(x_{3} > \\frac{1}{2}\\left(x_{1} + x_{2}\\right)\\).\n\nFind ... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 29522,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \n$$\n\\frac{1}{1+\\frac{1}{1+\\frac{1}{1+\\ddots-\\frac{1}{1}}}}=\\frac{m}{n}\n$$\nwhere \\(m\\) and \\(n\\) are coprime natural numbers, and there are 1988 fraction lines on the left-hand side of the equation. Calculate the value of \\(m^2 + mn - n^2\\). Let's think step ... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 29523,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a positive integer $n$, let $\\theta(n)$ denote the number of integers $0 \\leq x<2010$ such that $x^{2}-n$ is divisible by 2010. Determine the remainder when $\\sum_{n=0}^{2009} n \\cdot \\theta(n)$ is divided by 2010. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "335"
} | {
"split": "train",
"index": 29524,
"solution": "Let us consider the $\\operatorname{sum} \\sum_{n=0}^{2009} n \\cdot \\theta(n)(\\bmod 2010)$ in a another way. Consider the sum $0^{2}+1^{2}+2^{2}+\\cdots+2007^{2}(\\bmod 2010)$. For each $0 \\leq n<2010$, in the latter sum, the term $n$ appears $\\theta(n)$ times... |
deepscaler | [
{
"role": "user",
"content": "Let $2005 = c_1 \\cdot 3^{a_1} + c_2 \\cdot 3^{a_2} + \\ldots + c_n \\cdot 3^{a_n}$, where $n$ is a positive integer, $a_1, a_2, \\ldots, a_n$ are distinct natural numbers (including 0, with the convention that $3^0 = 1$), and each of $c_1, c_2, \\ldots, c_n$ is equal to 1 or -... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 29525,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If point \\( P \\) is on the curve \\( y=\\frac{1}{2} e^{x} \\) and point \\( Q \\) is on the curve \\( y=\\ln (2 x) \\), then the minimum value of \\( |PQ| \\) is \\( \\qquad \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}(1 - \\ln 2)"
} | {
"split": "train",
"index": 29526,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that point $(x, y)$ moves on the circle $x^{2}+(y-1)^{2}=1$.\n(1) Find the maximum and minimum values of $\\frac{y-1}{x-2}$;\n(2) Find the maximum and minimum values of $2x+y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1 - \\sqrt{5}"
} | {
"split": "train",
"index": 29527,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The side of a triangle are 2, $\\sqrt{6},$ and $1 + \\sqrt{3}.$ Enter the angles of the triangle in degrees, separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75^\\circ"
} | {
"split": "train",
"index": 29528,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $F_n$ be the $n$th Fibonacci number, where as usual $F_1 = F_2 = 1$ and $F_{n + 1} = F_n + F_{n - 1}.$ Then\n\\[\\prod_{k = 2}^{100} \\left( \\frac{F_k}{F_{k - 1}} - \\frac{F_k}{F_{k + 1}} \\right) = \\frac{F_a}{F_b}\\]for some positive integers $a$ and $b.$ Enter the ord... | math | {
"style": "rule",
"ground_truth": "(100,101)"
} | {
"split": "train",
"index": 29529,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the ray $(0,+\\infty)$ of the number line, there are several (more than two) segments of length 1. For any two different segments, you can select one number from each so that these numbers differ exactly by a factor of 2. The left end of the leftmost segment is the number $a... | math | {
"style": "rule",
"ground_truth": "5.5"
} | {
"split": "train",
"index": 29530,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is infinite sequence of composite numbers $a_1,a_2,...,$ where $a_{n+1}=a_n-p_n+\\frac{a_n}{p_n}$ ; $p_n$ is smallest prime divisor of $a_n$ . It is known, that $37|a_n$ for every $n$ .\nFind possible values of $a_1$ Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "37^2"
} | {
"split": "train",
"index": 29531,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest integer $n$ such that $(x^2+y^2+z^2)^2 \\le n(x^4+y^4+z^4)$ for all real numbers $x,y$, and $z$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 29532,
"solution": "To find the smallest integer $n$ such that $(x^2+y^2+z^2)^2\\le n(x^4+y^4+z^4)$ holds for all real numbers $x, y, z$, we can utilize the Cauchy-Schwarz inequality in its sum of squares form.\n\n1. **Applying Cauchy-Schwarz Inequality:**\n The Cauchy-Schwarz inequ... |
deepscaler | [
{
"role": "user",
"content": "A paperboy delivers newspapers to 10 houses along Main Street. Wishing to save effort, he doesn't always deliver to every house, but to avoid being fired he never misses three consecutive houses. Compute the number of ways the paperboy could deliver papers in this manner.\n Let... | math | {
"style": "rule",
"ground_truth": "504"
} | {
"split": "train",
"index": 29533,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $M$ and $N$ are the midpoints of sides $PA$ and $PB$ of $\\triangle PAB$. As $P$ moves along a line that is parallel to side $AB$, how many of the four quantities listed below change?\n(a) the length of the segment $MN$\n(b) the perimeter of $\\triangle PAB$\n(c) the are... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 29534,
"solution": "To determine how many of the four quantities change as $P$ moves along a line parallel to side $AB$, we analyze each quantity individually:\n\n1. **Length of segment $MN$:**\n - Since $M$ and $N$ are midpoints of $PA$ and $PB$ respectively, segment $MN$ is parall... |
deepscaler | [
{
"role": "user",
"content": "Find the number of natural numbers not exceeding 2022 and not belonging to either the arithmetic progression \\(1, 3, 5, \\ldots\\) or the arithmetic progression \\(1, 4, 7, \\ldots\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "674"
} | {
"split": "train",
"index": 29535,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To investigate a non-luminous black planet in distant space, Xiao Feitian drives a high-speed spaceship equipped with a powerful light, traveling straight towards the black planet at a speed of 100,000 km/s. When Xiao Feitian had just been traveling for 100 seconds, the spacesh... | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"split": "train",
"index": 29536,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\omega$ be a nonreal root of $z^3 = 1.$ Let $a_1,$ $a_2,$ $\\dots,$ $a_n$ be real numbers such that\n\\[\\frac{1}{a_1 + \\omega} + \\frac{1}{a_2 + \\omega} + \\dots + \\frac{1}{a_n + \\omega} = 2 + 5i.\\]Compute\n\\[\\frac{2a_1 - 1}{a_1^2 - a_1 + 1} + \\frac{2a_2 - 1}{a_... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 29537,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "During a class's May Day gathering, the original program schedule included 5 events. Just before the performance, 2 additional events were added. If these 2 new events are to be inserted into the original schedule, how many different insertion methods are there? Let's think ste... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 29538,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues ... | math | {
"style": "rule",
"ground_truth": "103"
} | {
"split": "train",
"index": 29539,
"solution": "1. **Define the variables and equations:**\n Let $x$ be the number of times Alex visits the first booth, and $y$ be the number of times he visits the second booth. Each visit to the first booth changes the token counts as follows: Alex loses 2 red tokens and gain... |
deepscaler | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ composed of positive numbers, with a common ratio $q=2$, and $a_1 \\cdot a_2 \\cdot a_3 \\cdot \\ldots \\cdot a_{30} = 2^{45}$, calculate the product of the terms $a_1 \\cdot a_4 \\cdot a_7 \\cdot \\ldots \\cdot a_{28}$. Let's think step b... | math | {
"style": "rule",
"ground_truth": "2^5"
} | {
"split": "train",
"index": 29540,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At the beginning of school year in one of the first grade classes: $i)$ every student had exatly $20$ acquaintances $ii)$ every two students knowing each other had exactly $13$ mutual acquaintances $iii)$ every two students not knowing each other had exactly $12$ mutua... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 29541,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\begin{vmatrix} 2 & 0 & -1 \\\\ 7 & 4 & -3 \\\\ 2 & 2 & 5 \\end{vmatrix}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 29542,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of each of Art's cookies is given by the formula for the area of a trapezoid: A = 1/2h(b1 + b2). The total area of dough used by Art to make a batch of cookies is 240 square inches, and this is equivalent to the number of cookies multiplied by the area of each cookie. ... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 29543,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Travis is hopping around on the vertices of a cube. Each minute he hops from the vertex he's currently on to the other vertex of an edge that he is next to. After four minutes, what is the probability that he is back where he started? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "7/27"
} | {
"split": "train",
"index": 29544,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $XYZ$, $XY = 540$ and $YZ = 360$. Points $N$ and $O$ are located on $\\overline{XY}$ and $\\overline{XZ}$ respectively, such that $XN = NY$, and $\\overline{ZO}$ is the angle bisector of angle $Z$. Let $Q$ be the point of intersection of $\\overline{YN}$ and $\\over... | math | {
"style": "rule",
"ground_truth": "216"
} | {
"split": "train",
"index": 29545,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find a constant \\( A > 0 \\) such that for all real \\( x \\geqslant A \\), we have\n\n\\[ x^{3} + x + 1 \\leq \\frac{x^{4}}{1000000} \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1000000"
} | {
"split": "train",
"index": 29546,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Lisa has 10 friends and 34 marbles. What is the minimum number of additional marbles she needs so that she can give each friend at least one marble and no two friends receive the same number of marbles? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 29547,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A space probe travels $5,555,555,555,555$ kilometers in its journey towards a distant star. After reaching its destination, it has traveled $3,333,333,333,333$ kilometers from its last refueling point. How many kilometers did the probe travel before the last refueling point?\nA... | math | {
"style": "rule",
"ground_truth": "2,222,222,222,222"
} | {
"split": "train",
"index": 29548,
"solution": ""
} |
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