data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Let $x_{1}, x_{2}, \\ldots, x_{2022}$ be nonzero real numbers. Suppose that $x_{k}+\\frac{1}{x_{k+1}}<0$ for each $1 \\leq k \\leq 2022$, where $x_{2023}=x_{1}$. Compute the maximum possible number of integers $1 \\leq n \\leq 2022$ such that $x_{n}>0$. Let's think step by step... | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"split": "train",
"index": 30449,
"solution": "Let the answer be $M$. If $M>1011$, there would exist two consecutive positive terms $x_{k}, x_{k+1}$ which contradicts the assumption that $x_{k}+\\frac{1}{x_{k+1}}<0$. Thus, $M \\leq 1011$. If $M=1011$, then the $2022 x_{i}$ s must alternate between positive and ... |
deepscaler | [
{
"role": "user",
"content": "What is the probability that a randomly drawn positive factor of $60$ is less than $7$? 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": 30450,
"solution": "\nTo solve this problem, we need to determine the total number of positive factors of $60$ and how many of these factors are less than $7$. We then calculate the probability by dividing the number of favorable outcomes (factors less than $7$) by the total number of... |
deepscaler | [
{
"role": "user",
"content": "Given the fraction \\(\\frac{5}{1+\\sqrt[3]{32 \\cos ^{4} 15^{\\circ}-10-8 \\sqrt{3}}}\\). Simplify the expression under the cubic root to a simpler form, and then reduce the fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1 - \\sqrt[3]{4} + \\sqrt[3]{16}"
} | {
"split": "train",
"index": 30451,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In his spare time, Thomas likes making rectangular windows. He builds windows by taking four $30\\text{ cm}\\times20\\text{ cm}$ rectangles of glass and arranging them in a larger rectangle in wood. The window has an $x\\text{ cm}$ wide strip of wood between adjacent glass ... | math | {
"style": "rule",
"ground_truth": "\\frac{20}{3}"
} | {
"split": "train",
"index": 30452,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all triples $(a, b, c)$ of positive integers such that $a^3 + b^3 + c^3 = (abc)^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(3, 2, 1), (3, 1, 2), (2, 3, 1), (2, 1, 3), (1, 3, 2), (1, 2, 3)"
} | {
"split": "train",
"index": 30453,
"solution": "\nWe are required to find all triples \\((a, b, c)\\) of positive integers such that:\n\n\\[\na^3 + b^3 + c^3 = (abc)^2\n\\]\n\n### Step 1: Analyze the Equation\n\nWe start by analyzing the given equation:\n\n\\[\na^3 + b^3 + c^3 = (abc)^2\n\\]\n\nWe will assume, w... |
deepscaler | [
{
"role": "user",
"content": "Jitka hiked a trail. After hiking 60% of the length of the trail, she had 8 km left to go. What is the length of the trail? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20 \\text{ km}"
} | {
"split": "train",
"index": 30454,
"solution": "After Jitka hiked 60% of the trail, 40% of the trail was left, which corresponds to 8 km. This means that 10% of the trail corresponds to 2 km. Therefore, the total length of the trail is \\( 10 \\times 2 = 20 \\text{ km} \\)."
} |
deepscaler | [
{
"role": "user",
"content": "What is the modulo $5$ remainder of the sum $1+2+3+4+5+ \\ldots + 120+121+122+123?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 30455,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When a class of math students lined up for a class picture on Picture Day, they found that when they stood in rows of four there was one person left over. When they formed rows of five students, there were two extra students, and when they lined up in rows of seven, there were ... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 30456,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The lines $y=2$, $y=5$, $x=1$, and $x=a$ make a square. Find the product of the possible values for $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"split": "train",
"index": 30457,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Either increasing the radius or the height of a cylinder by six inches will result in the same volume. The original height of the cylinder is two inches. What is the original radius in inches? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 30458,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The measures of the interior angles of a convex hexagon form an increasing arithmetic sequence. How many such sequences are possible if the hexagon is not equiangular and all of the angle degree measures are positive integers less than $150$ degrees? Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 30459,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the parallelogram \\(ABCD\\), points \\(E\\) and \\(F\\) are located on sides \\(AB\\) and \\(BC\\) respectively, and \\(M\\) is the point of intersection of lines \\(AF\\) and \\(DE\\). Given that \\(AE = 2BE\\) and \\(BF = 3CF\\), find the ratio \\(AM : MF\\). Let's think ... | math | {
"style": "rule",
"ground_truth": "4:5"
} | {
"split": "train",
"index": 30460,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Shaq sees the numbers $1$ through $2017$ written on a chalkboard. He repeatedly chooses three numbers, erases them, and writes one plus their median. (For instance, if he erased $-2, -1, 0$ he would replace them with $0$ .) If $M$ is the maximum possible final value re... | math | {
"style": "rule",
"ground_truth": "2014"
} | {
"split": "train",
"index": 30461,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $AB = 10$ and $AC = 17$. Let $D$ be the foot of the perpendicular from $A$ to $BC$. If $BD:CD = 2:5$, then find $AD$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 30462,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)$ defined on $\\mathbb{R}$ and satisfying the condition $f(x+2) = 3f(x)$, when $x \\in [0, 2]$, $f(x) = x^2 - 2x$. Find the minimum value of $f(x)$ when $x \\in [-4, -2]$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{9}"
} | {
"split": "train",
"index": 30463,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $a$, $b$, $c$ are the lengths of the sides opposite to $\\angle A$, $\\angle B$, $\\angle C$ respectively. It is known that $a$, $b$, $c$ are in geometric progression, and $a^{2}-c^{2}=ac-bc$,\n(1) Find the measure of $\\angle A$;\n(2) Find the value of $\\... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 30464,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two positive integers differ by $60$. The sum of their square roots is the square root of an integer that is not a perfect square. What is the maximum possible sum of the two integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "156"
} | {
"split": "train",
"index": 30465,
"solution": "Call the two integers $b$ and $b+60$, so we have $\\sqrt{b}+\\sqrt{b+60}=\\sqrt{c}$. Square both sides to get $2b+60+2\\sqrt{b^2+60b}=c$. Thus, $b^2+60b$ must be a square, so we have $b^2+60b=n^2$, and $(b+n+30)(b-n+30)=900$. The sum of these two factors is $2b+60$... |
deepscaler | [
{
"role": "user",
"content": "For all triples \\((x, y, z)\\) that satisfy the system\n\n$$\n\\left\\{\\begin{array}{l}\n2 \\cos x = \\operatorname{ctg} y \\\\\n2 \\sin y = \\operatorname{tg} z \\\\\n\\cos z = \\operatorname{ctg} x\n\\end{array}\\right.\n$$\n\nfind the minimum value of the expression \\(\\s... | math | {
"style": "rule",
"ground_truth": "-\\frac{5 \\sqrt{3}}{6}"
} | {
"split": "train",
"index": 30466,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the ellipse $\\dfrac {x^{2}}{36}+ \\dfrac {y^{2}}{9}=1$, there are two moving points $M$ and $N$, and $K(2,0)$ is a fixed point. If $\\overrightarrow{KM} \\cdot \\overrightarrow{KN} = 0$, find the minimum value of $\\overrightarrow{KM} \\cdot \\overrightarrow{NM}$. Let's thi... | math | {
"style": "rule",
"ground_truth": "\\dfrac{23}{3}"
} | {
"split": "train",
"index": 30467,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "You have a rectangular prism box with length $x+5$ units, width $x-5$ units, and height $x^{2}+25$ units. For how many positive integer values of $x$ is the volume of the box less than 700 units? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 30468,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two distinct integers $x$ and $y$ are factors of 48. One of these integers must be even. If $x\\cdot y$ is not a factor of 48, what is the smallest possible value of $x\\cdot y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 30469,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a right rectangular prism $B$ with edge lengths $1,$ $3,$ and $4$, and the set $S(r)$ of points within a distance $r$ of some point in $B$, express the volume of $S(r)$ as $ar^{3} + br^{2} + cr + d$ and determine the ratio $\\frac{bc}{ad}$. Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 30470,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At Barker High School, a total of 36 students are on either the baseball team, the hockey team, or both. If there are 25 students on the baseball team and 19 students on the hockey team, how many students play both sports? Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 30471,
"solution": "The two teams include a total of $25+19=44$ players. There are exactly 36 students who are on at least one team. Thus, there are $44-36=8$ students who are counted twice. Therefore, there are 8 students who play both baseball and hockey."
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a convex quadrilateral with $AB = CD = 10$, $BC = 14$, and $AD = 2\\sqrt{65}$. Assume that the diagonals of $ABCD$ intersect at point $P$, and that the sum of the areas of triangles $APB$ and $CPD$ equals the sum of the areas of triangles $BPC$ and $APD$. Find the... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 30472,
"solution": "1. **Identify the Relationship Between Angles and Sides**:\n Given that $\\angle APB = \\angle CPD$ and $\\angle APD = \\angle BPC$, and using the property that $\\sin(\\theta) = \\sin(180^\\circ - \\theta)$, we can apply the sine area formula for triangles. This... |
deepscaler | [
{
"role": "user",
"content": "Four steel balls, each with a radius of 1, are completely packed into a container in the shape of a regular tetrahedron. The minimum height of this regular tetrahedron is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 + \\frac{2 \\sqrt{6}}{3}"
} | {
"split": "train",
"index": 30473,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of pairs of integers \\((x ; y)\\) that satisfy the equation \\(y^{2} - xy = 700000000\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "324"
} | {
"split": "train",
"index": 30474,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $M$ such that both $M$ and $M^2$ end in the same sequence of three digits $xyz$ when written in base $10$, where $x$ is not zero. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "376"
} | {
"split": "train",
"index": 30475,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $O$ be the origin. Determine the scalar $m$ such that for any points $A, B, C, D$, if the equation\n\\[4 \\overrightarrow{OA} - 3 \\overrightarrow{OB} + 6 \\overrightarrow{OC} + m \\overrightarrow{OD} = \\mathbf{0}\\]\nholds, then the points $A, B, C$, and $D$ are coplanar.... | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"split": "train",
"index": 30476,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the set \\(I = \\{0, 1, 2, \\ldots, 22\\}\\). Define \\(A = \\{(a, b, c, d) \\mid a, b, c, d \\in I, a + d \\equiv 1 \\pmod{23}, \\text{ and } a d - b c \\equiv 0 \\pmod{23}\\}\\). Determine the number of elements in the set \\(A\\). Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "552"
} | {
"split": "train",
"index": 30477,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given complex numbers \\( z \\) and \\( \\omega \\) that satisfy the following two conditions:\n1. \\( z + \\omega + 3 = 0 \\);\n2. \\( | z |, 2, | \\omega | \\) form an arithmetic sequence.\nDoes \\( \\cos (\\arg z - \\arg \\omega) \\) have a maximum value? If it does, find th... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 30478,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C$ be a triangle and $D, E$, and $F$ be the midpoints of sides $B C, C A$, and $A B$ respectively. What is the maximum number of circles which pass through at least 3 of these 6 points? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 30479,
"solution": "All $\\binom{6}{3}=20$ triples of points can produce distinct circles aside from the case where the three points are collinear $(B D C, C E A, A F B)$."
} |
deepscaler | [
{
"role": "user",
"content": "For certain real numbers $a$, $b$, and $c$, the polynomial \\[g(x) = x^3 + ax^2 + 2x + 15\\] has three distinct roots, which are also roots of the polynomial \\[f(x) = x^4 + x^3 + bx^2 + 75x + c.\\] Determine the value of $f(-1)$. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "-2773"
} | {
"split": "train",
"index": 30480,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a function $f(x)$ on $\\mathbb{R}$ that satisfies $f(x+6)=f(x)$. When $x \\in [-3,-1)$, $f(x)=-(x+2)^{2}$, and when $x \\in [-1,3)$, $f(x)=x$. Find the value of $f(1)+f(2)+f(3)+\\ldots+f(2016)$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "336"
} | {
"split": "train",
"index": 30481,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rubber ball is released from a height of 120 feet and rebounds to three-quarters of the height it falls each time it bounces. How far has the ball traveled when it strikes the ground for the fifth time? Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "612.1875"
} | {
"split": "train",
"index": 30482,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rectangle $PQRS,$ $PQ=12$ and $PR=13.$ What is the area of rectangle $PQRS?$\n\n[asy]\ndraw((0,0)--(12,0)--(12,5)--(0,5)--cycle,black+linewidth(1));\ndraw((0,5)--(12,0),black+linewidth(1));\nlabel(\"$P$\",(0,5),NW);\nlabel(\"$Q$\",(12,5),NE);\nlabel(\"$R$\",(12,0),SE);\nlabe... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 30483,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is known that the probabilities of person A and person B hitting the target in each shot are $\\frac{3}{4}$ and $\\frac{4}{5}$, respectively. Person A and person B do not affect each other's chances of hitting the target, and each shot is independent. If they take turns shoo... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{100}"
} | {
"split": "train",
"index": 30484,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\\frac{n}{180}$ terminate? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 30485,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\sqrt[4]{256000000}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40\\sqrt{10}"
} | {
"split": "train",
"index": 30486,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A stack of logs has 12 logs on the bottom row, and one less in each successive row, ending with three logs at the top. How many logs are in the stack? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 30487,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A$ and $B$ be the endpoints of a semicircular arc of radius $2$. The arc is divided into seven congruent arcs by six equally spaced points $C_1$, $C_2$, $\\dots$, $C_6$. All chords of the form $\\overline {AC_i}$ or $\\overline {BC_i}$ are drawn. Find the product of the le... | math | {
"style": "rule",
"ground_truth": "28672"
} | {
"split": "train",
"index": 30488,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers less than $800$ are either a perfect cube or a perfect square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 30489,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) : \\mathbb{R} \\to \\mathbb{R}$ be a function such that\n\\[\\frac{f(x) f(y) - f(xy)}{3} = x + y + 2\\]for all $x,$ $y \\in \\mathbb{R}.$ Find $f(x).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x + 3"
} | {
"split": "train",
"index": 30490,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each morning of her five-day workweek, Jane bought either a $50$-cent muffin or a $75$-cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 30491,
"solution": "Let $b$ be the number of bagels Jane bought, and $m$ be the number of muffins. Since Jane buys these items over a five-day workweek, we have the equation:\n\\[ b + m = 5 \\]\n\nThe cost of a bagel is $75$ cents and the cost of a muffin is $50$ cents. The total cost... |
deepscaler | [
{
"role": "user",
"content": "On the base \\( AC \\) of an isosceles triangle \\( ABC \\), a circle is constructed using it as a diameter. This circle intersects side \\( BC \\) at point \\( N \\) such that \\( BN: NC = 7: 2 \\). Find the ratio of the lengths of segments \\( AN \\) and \\( BC \\). Let's thi... | math | {
"style": "rule",
"ground_truth": "\\frac{4 \\sqrt{2}}{9}"
} | {
"split": "train",
"index": 30492,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Starting with some gold coins and some empty treasure chests, I tried to put $9$ gold coins in each treasure chest, but that left $2$ treasure chests empty. So instead I put $6$ gold coins in each treasure chest, but then I had $3$ gold coins left over. How many gold coins di... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 30493,
"solution": "Let $n$ be the number of gold coins and $c$ be the number of treasure chests.\n\n1. **Analyzing the first condition:**\n - If we try to put $9$ gold coins in each chest, but $2$ chests remain empty, then the number of chests that actually contain coins is $c - 2$... |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathcal{F}$ be the set of continuous functions $f: \\mathbb{R} \\to \\mathbb{R}$ such that $$ e^{f(x)}+f(x) \\geq x+1, \\: \\forall x \\in \\mathbb{R} $$ For $f \\in \\mathcal{F},$ let $$ I(f)=\\int_0^ef(x) dx $$ Determine $\\min_{f \\in \\mathcal{F}}I(f).$ ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 30494,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the set $I=\\{1,2,3,4,5\\}$. Choose two non-empty subsets $A$ and $B$ from $I$ such that the smallest number in $B$ is greater than the largest number in $A$. The number of different ways to choose such subsets $A$ and $B$ is ______. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 30495,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The greatest common divisor of 30 and some number between 70 and 90 is 6. What is the number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "78"
} | {
"split": "train",
"index": 30496,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A segment of length $1$ is divided into four segments. Then there exists a quadrilateral with the four segments as sides if and only if each segment is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x < \\frac{1}{2}"
} | {
"split": "train",
"index": 30497,
"solution": "To determine the conditions under which four segments can form a quadrilateral, we need to consider the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For four ... |
deepscaler | [
{
"role": "user",
"content": "Points $F$, $G$, $H$, $I$, and $J$ are positioned in 3-dimensional space with $FG = GH = HI = IJ = JF = 3$ and $\\angle FGH = \\angle HIJ = \\angle JIF = 90^\\circ$. The plane of triangle $FGH$ is parallel to $\\overline{IJ}$. Determine the area of triangle $GIJ$. Let's think s... | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"split": "train",
"index": 30498,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle has vertices $A(0,0)$, $B(12,0)$, and $C(8,10)$. The probability that a randomly chosen point inside the triangle is closer to vertex $B$ than to either vertex $A$ or vertex $C$ can be written as $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integer... | math | {
"style": "rule",
"ground_truth": "409"
} | {
"split": "train",
"index": 30499,
"solution": "Calculate the area of the triangle using the Shoelace Theorem on $(0,0), (12,0), (8,10)$ \\[\\frac{1}{2}|(0+120+0)-(0+0+0)|=60\\]\n Get the four points $(6,0), (6,\\frac{17}{5}), (10,5)$, and $(12,0)$ by any method from the above solutions. Then use the Shoelace Th... |
deepscaler | [
{
"role": "user",
"content": "Given that \\(a, b, c, a+b-c, a+c-b, b+c-a, a+b+c\\) are 7 distinct prime numbers, and the sum of any two of \\(a, b, c\\) is 800. Let \\(d\\) be the difference between the largest and smallest of these 7 prime numbers. Find the maximum possible value of \\(d\\). Let's think st... | math | {
"style": "rule",
"ground_truth": "1594"
} | {
"split": "train",
"index": 30500,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The bases \\(AB\\) and \\(CD\\) of trapezoid \\(ABCD\\) are 55 and 31 respectively, and its diagonals are mutually perpendicular. Find the dot product of vectors \\(\\overrightarrow{AD}\\) and \\(\\overrightarrow{BC}\\). Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "1705"
} | {
"split": "train",
"index": 30501,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate \\[ \\lim_{x \\to 1^-} \\prod_{n=0}^\\infty \\left(\\frac{1 + x^{n+1}}{1 + x^n}\\right)^{x^n}. \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{e}"
} | {
"split": "train",
"index": 30502,
"solution": "By taking logarithms, we see that the desired limit is $\\exp(L)$, where $L = \\lim_{x\\to 1^-} \\sum_{n=0}^{\\infty} x^n \\left( \\ln(1+x^{n+1}) - \\ln(1+x^n) \\right)$. Now \\begin{align*} &\\sum_{n=0}^N x^n \\left( \\ln(1+x^{n+1}) - \\ln(1+x^n) \\right) \\\\ & =... |
deepscaler | [
{
"role": "user",
"content": "The energy stored by any pair of positive charges is inversely proportional to the distance between them, and directly proportional to their charges. Three identical point charges start at the vertices of an equilateral triangle, and this configuration stores 15 Joules of energ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 30503,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain college student had the night of February 23 to work on a chemistry problem set and a math problem set (both due on February 24, 2006). If the student worked on his problem sets in the math library, the probability of him finishing his math problem set that night is 9... | math | {
"style": "rule",
"ground_truth": "95/159"
} | {
"split": "train",
"index": 30504,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The increasing sequence $1,3,4,9,10,12,13\\cdots$ consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Determine the $150^{\\mbox{th}}$ term of this sequence. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "2280"
} | {
"split": "train",
"index": 30505,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three squares \\( GQOP, HJNO \\), and \\( RKMN \\) have vertices which sit on the sides of triangle \\( FIL \\) as shown. The squares have areas of 10, 90, and 40 respectively. What is the area of triangle \\( FIL \\)? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "220.5"
} | {
"split": "train",
"index": 30506,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Emilia wishes to create a basic solution with $7 \\%$ hydroxide $(\\mathrm{OH})$ ions. She has three solutions of different bases available: $10 \\%$ rubidium hydroxide $(\\mathrm{Rb}(\\mathrm{OH})$ ), $8 \\%$ cesium hydroxide $(\\mathrm{Cs}(\\mathrm{OH})$ ), and $5 \\%$ franci... | math | {
"style": "rule",
"ground_truth": "1 \\%"
} | {
"split": "train",
"index": 30507,
"solution": "Suppose that Emilia uses $R$ liters of $\\mathrm{Rb}(\\mathrm{OH}), C$ liters of $\\mathrm{Cs}(\\mathrm{OH})$, and $F$ liters of $\\mathrm{Fr}(\\mathrm{OH})$, then we have $$\\frac{10 \\% \\cdot R+8 \\% \\cdot C+5 \\% \\cdot F}{R+C+F}=7 \\% \\text { and } \\frac{5 ... |
deepscaler | [
{
"role": "user",
"content": "In rectangle $ABCD$, $AB=7$ and $BC=4$. Points $J$ and $K$ are on $\\overline{CD}$ such that $DJ = 2$ and $KC=3$. Lines $AJ$ and $BK$ intersect at point $L$. Find the area of $\\triangle ABL$. Visualize as follows:\n[asy]\npair A,B,C,D,L,J,K;\nA=(0,0);\nB=(7,0);\nC=(7,4);\nD=(0... | math | {
"style": "rule",
"ground_truth": "\\frac{98}{5}"
} | {
"split": "train",
"index": 30508,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $0.25 \\cdot 0.08$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.02"
} | {
"split": "train",
"index": 30509,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The first term of a sequence is $2005$. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the ${2005}^{\\text{th}}$ term of the sequence? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "250"
} | {
"split": "train",
"index": 30510,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain organism starts with 4 cells. Each cell splits into two cells at the end of three days. However, at the end of each 3-day period, 10% of the cells die immediately after splitting. This process continues for a total of 9 days. How many cells are there at the end of the... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 30511,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If four coins are tossed at the same time, what is the probability of getting exactly three heads and one tail? Follow this by rolling a six-sided die. What is the probability that the die shows a number greater than 4? Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"split": "train",
"index": 30512,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A polynomial $p(x)$ leaves a remainder of $2$ when divided by $x - 3,$ a remainder of 1 when divided by $x - 4,$ and a remainder of 5 when divided by $x + 4.$ Let $r(x)$ be the remainder when $p(x)$ is divided by $(x - 3)(x - 4)(x + 4).$ Find $r(5).$ Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{13}"
} | {
"split": "train",
"index": 30513,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For the ellipse shown below, find the distance between the foci.\n\n[asy]\nunitsize(0.3 cm);\n\nint i, n = 10;\n\nfor (i = -n; i <= n; ++i) {\n draw((i,-n)--(i,n),gray(0.7));\n draw((-n,i)--(n,i),gray(0.7));\n}\n\ndraw((0,-n)--(0,n));\ndraw((-n,0)--(n,0));\n\ndraw(shift((1,1)... | math | {
"style": "rule",
"ground_truth": "8 \\sqrt{2}"
} | {
"split": "train",
"index": 30514,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A school is arranging for 5 trainee teachers, including Xiao Li, to be placed in Class 1, Class 2, and Class 3 for teaching practice. If at least one teacher must be assigned to each class and Xiao Li is to be placed in Class 1, the number of different arrangement schemes is __... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 30515,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $$\\lceil\\sqrt{5}\\rceil + \\lceil\\sqrt{6}\\rceil + \\lceil\\sqrt{7}\\rceil + \\cdots + \\lceil\\sqrt{29}\\rceil$$Note: For a real number $x,$ $\\lceil x \\rceil$ denotes the smallest integer that is greater than or equal to $x.$ Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "112"
} | {
"split": "train",
"index": 30516,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $\\frac{12}{15} + \\frac{7}{9} + 1\\frac{1}{6}$ and simplify the result to its lowest terms. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{247}{90}"
} | {
"split": "train",
"index": 30517,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the difference between the greatest and least values of $lcm (a,b,c)$ , where $a$ , $b$ , and $c$ are distinct positive integers between $1$ and $10$ , inclusive. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "500"
} | {
"split": "train",
"index": 30518,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Over all real numbers $x$ and $y$, find the minimum possible value of $$ (x y)^{2}+(x+7)^{2}+(2 y+7)^{2} $$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 30519,
"solution": "Solution 1: Rewrite the given expression as $\\left(x^{2}+4\\right)\\left(1+y^{2}\\right)+14(x+2 y)+94$. By Cauchy-Schwartz, this is at least $(x+2 y)^{2}+14(x+2 y)+94=(x+2 y+7)^{2}+45$. The minimum is 45 , attained when $x y=2, x+2 y=-7$. Solution 2: Let $z=2 y, s... |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the greatest common divisor of $50$ and $5005$ and the least common multiple of $50$ and $5005$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50055"
} | {
"split": "train",
"index": 30520,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Small lights are hung on a string $6$ inches apart in the order red, red, green, green, green, red, red, green, green, green, and so on continuing this pattern of $2$ red lights followed by $3$ green lights. How many feet separate the 3rd red light and the 21st red light? Let's... | math | {
"style": "rule",
"ground_truth": "22.5"
} | {
"split": "train",
"index": 30521,
"solution": "1. **Identify the pattern and the position of red lights**: The pattern of lights is 2 red followed by 3 green. This pattern repeats every 5 lights. The red lights occur at positions 1, 2, 6, 7, 11, 12, and so on in each repeating group.\n\n2. **Determine the posit... |
deepscaler | [
{
"role": "user",
"content": "Consider a large square divided into a grid of \\(5 \\times 5\\) smaller squares, each with side length \\(1\\) unit. A shaded region within the large square is formed by connecting the centers of four smaller squares, creating a smaller square inside. Calculate the ratio of th... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{25}"
} | {
"split": "train",
"index": 30522,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p = 2027$ be the smallest prime greater than $2018$ , and let $P(X) = X^{2031}+X^{2030}+X^{2029}-X^5-10X^4-10X^3+2018X^2$ . Let $\\mathrm{GF}(p)$ be the integers modulo $p$ , and let $\\mathrm{GF}(p)(X)$ be the set of rational functions with coefficients in $\\ma... | math | {
"style": "rule",
"ground_truth": "4114810"
} | {
"split": "train",
"index": 30523,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the ten terms in the arithmetic sequence $-3, 4, \\dots, 40$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "285"
} | {
"split": "train",
"index": 30524,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $K$, $L$, $M$, and $N$ lie in the plane of the square $ABCD$ so that $AKB$, $BLC$, $CMD$, and $DNA$ are equilateral triangles. If $ABCD$ has an area of 16, find the area of $KLMN$. Express your answer in simplest radical form.\n\n[asy]\npair K,L,M,I,A,B,C,D;\nD=(0,0);\nC... | math | {
"style": "rule",
"ground_truth": "32 + 16\\sqrt{3}"
} | {
"split": "train",
"index": 30525,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate: $81^2 - (x+9)^2$ where $x=45$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3645"
} | {
"split": "train",
"index": 30526,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the lengths of the arcs of the curves given by the equations in the rectangular coordinate system.\n\n\\[ y = \\ln \\frac{5}{2 x}, \\quad \\sqrt{3} \\leq x \\leq \\sqrt{8} \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1 + \\frac{1}{2} \\ln \\frac{3}{2}"
} | {
"split": "train",
"index": 30527,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Owen spends $\\$ 1.20$ per litre on gasoline. He uses an average of 1 L of gasoline to drive 12.5 km. How much will Owen spend on gasoline to drive 50 km? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\$ 4.80"
} | {
"split": "train",
"index": 30528,
"solution": "Since Owen uses an average of 1 L to drive 12.5 km, then it costs Owen $\\$ 1.20$ in gas to drive 12.5 km. To drive 50 km, he drives 12.5 km a total of $50 \\div 12.5=4$ times. Therefore, it costs him $4 \\times \\$ 1.20=\\$ 4.80$ in gas to drive 50 km."
} |
deepscaler | [
{
"role": "user",
"content": "Given $cosθ+cos(θ+\\frac{π}{3})=\\frac{\\sqrt{3}}{3},θ∈(0,\\frac{π}{2})$, find $\\sin \\theta$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-1 + 2\\sqrt{6}}{6}"
} | {
"split": "train",
"index": 30529,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are several teacups in the kitchen, some with handles and the others without handles. The number of ways of selecting two cups without a handle and three with a handle is exactly $1200$ . What is the maximum possible number of cups in the kitchen? Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 30530,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The school plans to set up two computer labs, each equipped with one teacher's computer and several student computers. In a standard lab, the teacher's computer costs 8000 yuan, and each student computer costs 3500 yuan; in an advanced lab, the teacher's computer costs 11500 yu... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 30531,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( N = 34 \\times 34 \\times 63 \\times 270 \\). The ratio of the sum of all odd factors of \\( N \\) to the sum of all even factors of \\( N \\) is ( ). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1: 14"
} | {
"split": "train",
"index": 30532,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all pairs $(m,n)$ of nonnegative integers for which \\[m^2 + 2 \\cdot 3^n = m\\left(2^{n+1} - 1\\right).\\]\n\n[i] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(9, 3), (6, 3), (9, 5), (54, 5)"
} | {
"split": "train",
"index": 30533,
"solution": "\nWe are tasked with finding all pairs \\((m, n)\\) of nonnegative integers that satisfy the equation:\n\\[\nm^2 + 2 \\cdot 3^n = m\\left(2^{n+1} - 1\\right).\n\\]\n\nTo solve this equation, we rearrange terms to express it in a form that can be factored:\n\\[\nm^2... |
deepscaler | [
{
"role": "user",
"content": "In the expansion of $({x-1}){({\\frac{1}{{{x^{2022}}}}+\\sqrt{x}+1})^8}$, the coefficient of $x^{2}$ is _____. (Provide your answer as a number). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-42"
} | {
"split": "train",
"index": 30534,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)$ be the function such that $f(x)>0$ at $x\\geq 0$ and $\\{f(x)\\}^{2006}=\\int_{0}^{x}f(t) dt+1.$ \r\n\r\nFind the value of $\\{f(2006)\\}^{2005}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2006"
} | {
"split": "train",
"index": 30535,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N=2^{(2^{2})}$ and $x$ be a real number such that $N^{(N^{N})}=2^{(2^{x})}$. Find $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "66"
} | {
"split": "train",
"index": 30536,
"solution": "We compute $$N^{(N^{N})}=16^{16^{16}}=2^{4 \\cdot 2^{4 \\cdot 2^{4}}}=2^{2^{2^{6}+2}}=2^{2^{66}}$$ so $x=66$."
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram below, \\( \\triangle ABC \\) is a triangle with \\( AB = 39 \\text{ cm}, BC = 45 \\text{ cm}, \\) and \\( CA = 42 \\text{ cm} \\). The tangents at \\( A \\) and \\( B \\) to the circumcircle of \\( \\triangle ABC \\) meet at the point \\( P \\). The point \\( D ... | math | {
"style": "rule",
"ground_truth": "168"
} | {
"split": "train",
"index": 30537,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\overrightarrow{a} \\cdot \\overrightarrow{b} + 1$, where $\\overrightarrow{a} = (\\sqrt{3}, 2\\sin \\frac{\\omega x}{2})$ and $\\overrightarrow{b} = (\\sin \\omega x, -\\sin \\frac{\\omega x}{2})$, $\\omega > 0$, and the smallest positive period of ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{12}"
} | {
"split": "train",
"index": 30538,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that Ben constructs a $4$-step staircase using $26$ toothpicks, determine the number of additional toothpicks needed to extend the staircase to a $6$-step staircase, Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 30539,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sin 2α - 2 = 2 \\cos 2α$, find the value of ${\\sin}^{2}α + \\sin 2α$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{5}"
} | {
"split": "train",
"index": 30540,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N$ denote the sum of the decimal digits of $\\binom{1000}{100}$. Estimate the value of $N$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "621"
} | {
"split": "train",
"index": 30541,
"solution": "http://www.wolframalpha.com/input/?i=sum+of+digits+of $+\\mathrm{nCr}(1000,100)$. To see this, one can estimate there are about 150 digits, and we expect the digits to be roughly random, for $150 \\cdot 4.5 \\approx 675$, which is already very close to the actual a... |
deepscaler | [
{
"role": "user",
"content": "A rectangular box $Q$ is inscribed in a sphere of radius $s$. The surface area of $Q$ is 576, and the sum of the lengths of its 12 edges is 168. Determine the radius $s$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{33}"
} | {
"split": "train",
"index": 30542,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all functions $f: \\mathbb R \\to \\mathbb R$ such that for any $x,y \\in \\mathbb R$, the multiset $\\{(f(xf(y)+1),f(yf(x)-1)\\}$ is identical to the multiset $\\{xf(f(y))+1,yf(f(x))-1\\}$.\n\n[i]Note:[/i] The multiset $\\{a,b\\}$ is identical to the multiset $\\{c,d\\}$ ... | math | {
"style": "rule",
"ground_truth": "f(x) \\equiv x \\text{ or } f(x) \\equiv -x"
} | {
"split": "train",
"index": 30543,
"solution": "\nLet \\( f: \\mathbb{R} \\to \\mathbb{R} \\) be a function such that for any \\( x, y \\in \\mathbb{R} \\), the multiset \\( \\{ f(xf(y) + 1), f(yf(x) - 1) \\} \\) is identical to the multiset \\( \\{ xf(f(y)) + 1, yf(f(x)) - 1 \\} \\).\n\nWe aim to find all such ... |
deepscaler | [
{
"role": "user",
"content": "Seven students are standing in a line for a graduation photo. Among them, student A must be in the middle, and students B and C must stand together. How many different arrangements are possible? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "192"
} | {
"split": "train",
"index": 30544,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A dart board is shaped like a regular dodecagon divided into regions, with a square at its center. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?\n**A)** $\\frac{2+\\... | math | {
"style": "rule",
"ground_truth": "\\frac{2 - \\sqrt{3}}{6}"
} | {
"split": "train",
"index": 30545,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "You are given a number composed of three different non-zero digits, 7, 8, and a third digit which is not 7 or 8. Find the minimum value of the quotient of this number divided by the sum of its digits. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "11.125"
} | {
"split": "train",
"index": 30546,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\(x + \\frac{1}{y} = 3\\) and \\(y + \\frac{1}{z} = 3\\), what is the value of the product \\(xyz\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 30547,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For points P and Q on the curve $y = 1 - x^2$, which are situated on opposite sides of the y-axis, find the minimum area of the triangle formed by the tangents at P and Q and the x-axis. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8 \\sqrt{3}}{9}"
} | {
"split": "train",
"index": 30548,
"solution": ""
} |
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