data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "The difference between two positive integers is 12 and their product is 45. What is the sum of the integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 27149,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $P$ is on the ellipse $\\frac{x^{2}}{16}+ \\frac{y^{2}}{9}=1$. The maximum and minimum distances from point $P$ to the line $3x-4y=24$ are $\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{12(2- \\sqrt{2})}{5}"
} | {
"split": "train",
"index": 27150,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $x = \\frac{2}{3}$ and $y = \\frac{5}{2}$, find the value of $\\frac{1}{3}x^8y^9$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5^9}{2 \\cdot 3^9}"
} | {
"split": "train",
"index": 27151,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A four-digit number satisfies the following conditions:\n(1) If you simultaneously swap its unit digit with the hundred digit and the ten digit with the thousand digit, the value increases by 5940;\n(2) When divided by 9, the remainder is 8.\nFind the smallest odd four-digit nu... | math | {
"style": "rule",
"ground_truth": "1979"
} | {
"split": "train",
"index": 27152,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Nine nonnegative numbers have an average of 10. What is the greatest possible value for their median? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 27153,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a parallelepiped \\(A B C D A_{1} B_{1} C_{1} D_{1}\\), a point \\(X\\) is chosen on edge \\(A_{1} D_{1}\\), and a point \\(Y\\) is chosen on edge \\(B C\\). It is known that \\(A_{1} X = 5\\), \\(B Y = 3\\), and \\(B_{1} C_{1} = 14\\). The plane \\(C_{1} X Y\\) intersect... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 27154,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( P \\) be the parabola with equation \\( y = x^2 \\) and let \\( Q = (10, 6) \\). There are real numbers \\( r \\) and \\( s \\) such that the line through \\( Q \\) with slope \\( m \\) does not intersect \\( P \\) if and only if \\( r < m < s \\). What is \\( r + s \\)... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 27155,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( n \\) be the product of 3659893456789325678 and 342973489379256. Determine the number of digits of \\( n \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 27156,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the sum of the integers 122 and 78, express both numbers and the resulting sum in base-5. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1300_5"
} | {
"split": "train",
"index": 27157,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given point $P$ is a moving point on the ellipse $\\frac{x^2}{8} + \\frac{y^2}{4} = 1$ in the first quadrant, through point $P$, two tangents $PA$ and $PB$ are drawn to the circle $x^2 + y^2 = 4$, with the points of tangency being $A$ and $B$ respectively. The line $AB$ interse... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 27158,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a right triangle \\(ABC\\) with a right angle at \\(C\\), a circle is drawn with diameter \\(BC\\) of length 26. A tangent \\(AP\\) from point \\(A\\) to this circle (distinct from \\(AC\\)) is drawn. The perpendicular \\(PH\\) dropped onto segment \\(BC\\) intersects seg... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 27159,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a shelf, there are 4 different comic books, 5 different fairy tale books, and 3 different story books, all lined up in a row. If the fairy tale books cannot be separated from each other, and the comic books also cannot be separated from each other, how many different arrange... | math | {
"style": "rule",
"ground_truth": "345600"
} | {
"split": "train",
"index": 27160,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $x$, $y$, $z \\in \\mathbb{R}$, if $-1$, $x$, $y$, $z$, $-3$ form a geometric sequence, calculate the value of $xyz$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3\\sqrt{3}"
} | {
"split": "train",
"index": 27161,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A subscriber forgot the last digit of a phone number and therefore dials it randomly. What is the probability that they will have to dial the number no more than three times? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.3"
} | {
"split": "train",
"index": 27162,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\sum_{i=1}^{\\infty} \\frac{(i+1)(i+2)(i+3)}{(-2)^{i}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 27163,
"solution": "This is the power series of $\\frac{6}{(1+x)^{4}}$ expanded about $x=0$ and evaluated at $x=-\\frac{1}{2}$, so the solution is 96."
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $x^3 + x^2 + x + 1$ when $x = 3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 27164,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The taxicab distance between points $\\left(x_{1}, y_{1}\\right)$ and $\\left(x_{2}, y_{2}\\right)$ is $\\left|x_{2}-x_{1}\\right|+\\left|y_{2}-y_{1}\\right|$. A regular octagon is positioned in the $x y$ plane so that one of its sides has endpoints $(0,0)$ and $(1,0)$. Let $S$... | math | {
"style": "rule",
"ground_truth": "2309"
} | {
"split": "train",
"index": 27165,
"solution": "In the taxicab metric, the set of points that lie at most $d$ units away from some fixed point $P$ form a square centered at $P$ with vertices at a distance of $d$ from $P$ in directions parallel to the axes. The diagram above depicts the intersection of an octagon... |
deepscaler | [
{
"role": "user",
"content": "Given the parabola $C: y^2 = 2px \\ (0 < p < 4)$ with focus $F$, and a moving point $P$ on $C$. Let $A(4, 0)$ and $B(p, \\sqrt{2}p)$ be such that the minimum value of $|PA|$ is $\\sqrt{15}$. Find the value of $|BF|$. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"split": "train",
"index": 27166,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A monic polynomial of degree $n$, with real coefficients, has its first two terms after $x^n$ as $a_{n-1}x^{n-1}$ and $a_{n-2}x^{n-2}$. It’s known that $a_{n-1} = 2a_{n-2}$. Determine the absolute value of the lower bound for the sum of the squares of the roots of this polynomi... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 27167,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of all ordered triple of integers $(a_1,a_2,a_3)$ with $1 \\le a_1,a_2,a_3 \\le 10$. Each ordered triple in $S$ generates a sequence according to the rule $a_n=a_{n-1}\\cdot | a_{n-2}-a_{n-3} |$ for all $n\\ge 4$. Find the number of such sequences for which $... | math | {
"style": "rule",
"ground_truth": "494"
} | {
"split": "train",
"index": 27168,
"solution": "Let $a_1=x, a_2=y, a_3=z$. First note that if any absolute value equals 0, then $a_n=0$. Also note that if at any position, $a_n=a_{n-1}$, then $a_{n+2}=0$. Then, if any absolute value equals 1, then $a_n=0$. Therefore, if either $|y-x|$ or $|z-y|$ is less than or ... |
deepscaler | [
{
"role": "user",
"content": "Consider a central regular hexagon surrounded by six regular hexagons, each of side length $\\sqrt{2}$. Three of these surrounding hexagons are selected at random, and their centers are connected to form a triangle. Calculate the area of this triangle. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 27169,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that vertex $E$ of right $\\triangle ABE$ (with $\\angle ABE = 90^\\circ$) is inside square $ABCD$, and $F$ is the point of intersection of diagonal $BD$ and hypotenuse $AE$, find the area of $\\triangle ABF$ where the length of $AB$ is $\\sqrt{2}$. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 27170,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest integer \\( k \\) whose square \\( k^2 \\) is a factor of \\( 10! \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "720"
} | {
"split": "train",
"index": 27171,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many $3$-digit positive integers have digits whose product equals $36$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 27172,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Right triangle $ABC$ has leg lengths $AB=20$ and $BC=21$. Including $\\overline{AB}$ and $\\overline{BC}$, how many line segments with integer length can be drawn from vertex $B$ to a point on hypotenuse $\\overline{AC}$? Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 27173,
"solution": "1. **Identify the Triangle and Hypotenuse**: \n Given a right triangle $ABC$ with legs $AB = 20$ and $BC = 21$, we first calculate the length of the hypotenuse $AC$ using the Pythagorean theorem:\n \\[\n AC = \\sqrt{AB^2 + BC^2} = \\sqrt{20^2 + 21^2} = \\sqrt... |
deepscaler | [
{
"role": "user",
"content": "Given $\\alpha$ and $\\beta \\in \\left(0,\\pi \\right)$, where $\\tan \\alpha$ and $\\tan \\beta$ are two roots of the equation ${x^2}+3\\sqrt{3}x+4=0$, find the value of $\\alpha +\\beta$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4\\pi}{3}"
} | {
"split": "train",
"index": 27174,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $r$, $s$, and $t$ be solutions of the equation $x^3-5x^2+6x=9$.\n\nCompute $\\frac{rs}t + \\frac{st}r + \\frac{tr}s$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 27175,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\sin\\alpha= \\frac{1}{2}+\\cos\\alpha$, and $\\alpha\\in(0, \\frac{\\pi}{2})$, then $\\sin2\\alpha= \\_\\_\\_\\_\\_\\_$, $\\cos2\\alpha= \\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{7}}{4}"
} | {
"split": "train",
"index": 27176,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $X$ and $Y$ are inside a unit square. The score of a vertex of the square is the minimum distance from that vertex to $X$ or $Y$. What is the minimum possible sum of the scores of the vertices of the square? Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}+\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 27177,
"solution": "Let the square be $A B C D$. First, suppose that all four vertices are closer to $X$ than $Y$. Then, by the triangle inequality, the sum of the scores is $A X+B X+C X+D X \\geq A B+C D=2$. Similarly, suppose exactly two vertices are closer to $X$ than $Y$. Here, we... |
deepscaler | [
{
"role": "user",
"content": "In the diagram, the circle has center \\( O \\) and square \\( OPQR \\) has vertex \\( Q \\) on the circle. If the area of the circle is \\( 72 \\pi \\), the area of the square is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 27178,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the circle with center O, and diameters AB and CD where AB is perpendicular to CD, and chord DF intersects AB at E with DE = 6 and EF = 2, find the area of the circle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24\\pi"
} | {
"split": "train",
"index": 27179,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find \\( n \\) such that \\( 2^3 \\cdot 5 \\cdot n = 10! \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45360"
} | {
"split": "train",
"index": 27180,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sum of the first $n$ terms ($S_n$) of the sequence $\\{a_n\\}$ satisfies $S_n = 2a_n - 1$ ($n \\in \\mathbb{N}^*$).\n\n(1) Find the general term formula of the sequence $\\{a_n\\}$;\n\n(2) If the sequence $\\{b_n\\}$ satisfies $b_n = 1 + \\log_2 a_n$,\n\n(I) Find... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{3}"
} | {
"split": "train",
"index": 27181,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a triangle with side lengths $5$ , $4\\sqrt 2$ , and $7$ . What is the area of the triangle with side lengths $\\sin A$ , $\\sin B$ , and $\\sin C$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{25}"
} | {
"split": "train",
"index": 27182,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nTetrahedron \\(ABCD\\) has base \\( \\triangle ABC \\). Point \\( E \\) is the midpoint of \\( AB \\). Point \\( F \\) is on \\( AD \\) so that \\( FD = 2AF \\), point \\( G \\) is on \\( BD \\) so that \\( GD = 2BG \\), and point \\( H \\) is on \\( CD \\) so that \\( HD = 2... | math | {
"style": "rule",
"ground_truth": "1/10"
} | {
"split": "train",
"index": 27183,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the price savings of buying the computer at store A is $15 more than buying it at store B, and store A offers a 15% discount followed by a $90 rebate, while store B offers a 25% discount and no rebate, calculate the sticker price of the computer. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "750"
} | {
"split": "train",
"index": 27184,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A coordinate system is established with the origin as the pole and the positive half of the x-axis as the polar axis. Given the curve $C_1: (x-2)^2 + y^2 = 4$, point A has polar coordinates $(3\\sqrt{2}, \\frac{\\pi}{4})$, and the polar coordinate equation of line $l$ is $\\rho... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 27185,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all possible values of $s$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are \\[(\\cos 30^\\circ, \\sin 30^\\circ), (\\cos 45^\\circ, \\sin 45^\\circ), \\text{ and } (\\cos s^\\circ, \\sin s^\\circ)\\] is isosceles and its a... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 27186,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the polar equation of curve C is $\\rho - 6\\cos\\theta + 2\\sin\\theta + \\frac{1}{\\rho} = 0$, with the pole as the origin of the Cartesian coordinate system and the polar axis as the positive x-axis, establish a Cartesian coordinate system in the plane xOy. The line $l... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"split": "train",
"index": 27187,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The four points $A(-1,2), B(3,-4), C(5,-6),$ and $D(-2,8)$ lie in the coordinate plane. Compute the minimum possible value of $PA + PB + PC + PD$ over all points P . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 27188,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least positive integer value of $x$ such that $(3x)^2 + 3 \\cdot 29 \\cdot 3x + 29^2$ is a multiple of 43? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 27189,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The real sequence \\( x_0, x_1, x_2, \\ldots \\) is defined by \\( x_0 = 1 \\), \\( x_1 = 2 \\), and \\( n(n+1) x_{n+1} = n(n-1) x_n - (n-2) x_{n-1} \\). Find \\( \\frac{x_0}{x_1} + x_1 x_2 + \\cdots + \\frac{x_{50}}{x_{51}} \\). Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "1326"
} | {
"split": "train",
"index": 27190,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $h(x) = x - 3$ and $k(x) = 2x$. Also denote the inverses of these functions as $h^{-1}$ and $k^{-1}$. Compute:\n\\[ h(k^{-1}(h^{-1}(h^{-1}(k(h(28)))))) \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 27191,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Expand the product ${(x+3)(x-8)}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x^2 - 5x - 24"
} | {
"split": "train",
"index": 27192,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let positive numbers $x$ and $y$ satisfy: $x > y$, $x+2y=3$. Find the minimum value of $\\frac{1}{x-y} + \\frac{9}{x+5y}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{3}"
} | {
"split": "train",
"index": 27193,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $-7$ is a solution to $x^2 + bx -28 = 0$, what is the value of $b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 27194,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, $A+B=5C$, $\\sin \\left(A-C\\right)=2\\sin B$. \n$(1)$ Find $A$; \n$(2)$ If $CM=2\\sqrt{7}$ and $M$ is the midpoint of $AB$, find the area of $\\triangle ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"split": "train",
"index": 27195,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the drawing, there is a grid consisting of 25 small equilateral triangles.\n\nHow many rhombuses can be formed from two adjacent small triangles? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 27196,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $C$: $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$, where the upper vertex of $C$ is $A$, and the two foci are $F_{1}$ and $F_{2}$, with an eccentricity of $\\frac{1}{2}$. A line passing through $F_{1}$ and perpendicular to $AF_{2}$ intersects $C$ at po... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 27197,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The tangents of the angles of a triangle are in the ratio $1: 2: 3$. How do the sides of the triangle compare to each other in length? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5} : 2\\sqrt{2} : 3"
} | {
"split": "train",
"index": 27198,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least positive multiple of 21 that is greater than 380? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "399"
} | {
"split": "train",
"index": 27199,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $4\\times 4$ square array of dots, as in the figure below?\n[asy]size(2cm,2cm); for (int i=0; i<4; ++i) { for (int j=0; j<4; ++j) { filldraw(Circle((i, j), .05), blac... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 27200,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the median of the first ten positive integers? Express your answer as a decimal to the nearest tenth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5.5"
} | {
"split": "train",
"index": 27201,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number \\( N \\) has the smallest positive divisor 1, the second largest positive divisor \\( k \\), and the third largest positive divisor \\( m \\). Moreover, \\( k^k + m^m = N \\). What is \\( N \\)? Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "260"
} | {
"split": "train",
"index": 27202,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( A(2,0) \\) be a fixed point on the plane, \\( P\\left(\\sin \\left(2 t-60^{\\circ}\\right), \\cos \\left(2 t-60^{\\circ}\\right)\\right) \\) be a moving point. When \\( t \\) changes from \\( 15^{\\circ} \\) to \\( 45^{\\circ} \\), the area swept by the line segment \\(... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{6}"
} | {
"split": "train",
"index": 27203,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Francesca uses $100$ grams of lemon juice, $100$ grams of sugar, and $400$ grams of water to make lemonade. There are $25$ calories in $100$ grams of lemon juice and $386$ calories in $100$ grams of sugar. Water contains no calories. How many calories are in $200$ grams of her ... | math | {
"style": "rule",
"ground_truth": "137"
} | {
"split": "train",
"index": 27204,
"solution": "1. **Calculate Total Calories in the Lemonade Mixture**:\n Francesca uses $100$ grams of lemon juice, $100$ grams of sugar, and $400$ grams of water. The calorie content of each ingredient is given as:\n - Lemon juice: $25$ calories per $100$ grams\n - Sugar:... |
deepscaler | [
{
"role": "user",
"content": "In triangle $PQR$, we have $\\angle P = 90^\\circ$, $QR = 15$, and $\\tan R = 5\\cos Q$. What is $PQ$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6\\sqrt{6}"
} | {
"split": "train",
"index": 27205,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( M \\) be a subset of the set \\(\\{1, 2, 3, \\cdots, 15\\}\\), and suppose that the product of any three different elements in \\( M \\) is not a perfect square. Let \\( |M| \\) denote the number of elements in the set \\( M \\). Find the maximum value of \\( |M| \\). L... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 27206,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\( a + x^2 = 2015 \\), \\( b + x^2 = 2016 \\), \\( c + x^2 = 2017 \\), and \\( abc = 24 \\), find the value of \\( \\frac{a}{bc} + \\frac{b}{ac} + \\frac{c}{ab} - \\frac{1}{a} - \\frac{1}{b} - \\frac{1}{c} \\). Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "1/8"
} | {
"split": "train",
"index": 27207,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jenny and Jack run on a circular track. Jenny runs counterclockwise and completes a lap every 75 seconds, while Jack runs clockwise and completes a lap every 70 seconds. They start at the same place and at the same time. Between 15 minutes and 16 minutes from the start, a photo... | math | {
"style": "rule",
"ground_truth": "\\frac{23}{60}"
} | {
"split": "train",
"index": 27208,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The length of the hypotenuse of a right-angled triangle is 2cm longer than one of its legs, and the other leg is 6cm long. Find the length of its hypotenuse. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 27209,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A(2,0)$ be a fixed point in the plane, and let $P\\left(\\sin \\left(2 t-60^{\\circ}\\right), \\cos \\left(2 t-60^{\\circ}\\right)\\right)$ be a moving point. Find the area swept by the line segment $AP$ as $t$ changes from $15^{\\circ}$ to $45^{\\circ}$. Let's think step ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{6}"
} | {
"split": "train",
"index": 27210,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For positive integers $a$, $b$, and $c$ with $a < b < c$, consider collections of postage stamps in denominations $a$, $b$, and $c$ cents that contain at least one stamp of each denomination. If there exists such a collection that contains sub-collections worth every whole numb... | math | {
"style": "rule",
"ground_truth": "188"
} | {
"split": "train",
"index": 27211,
"solution": "Notice that we must have $a = 1$, otherwise $1$ cent stamp cannot be represented. At least $b-1$ numbers of $1$ cent stamps are needed to represent the values less than $b$. Using at most $c-1$ stamps of value $1$ and $b$, it can have all the values from $1$ to $c-... |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic-geometric sequence $\\{a_n\\}$ with the sum of its first $n$ terms denoted as $S_n$, if $a_3 - 4a_2 + 4a_1 = 0$, find the value of $\\frac{S_8}{S_4}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 27212,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A group with 7 young men and 7 young women was divided into pairs randomly. Find the probability that at least one pair consists of two women. Round the answer to two decimal places. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.96"
} | {
"split": "train",
"index": 27213,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of positive integer pairs $(a,b)$ that have $a$ dividing $b$ and $b$ dividing $2013^{2014}$ can be written as $2013n+k$ , where $n$ and $k$ are integers and $0\\leq k<2013$ . What is $k$ ? Recall $2013=3\\cdot 11\\cdot 61$ . Let's think step by st... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 27214,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of $y=\\frac{5x^2-9}{3x^2+5x+2}$ has vertical asymptotes at $x = a$ and $x = b$. Find $a + b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{5}{3}"
} | {
"split": "train",
"index": 27215,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find, with proof, the maximum positive integer \\(k\\) for which it is possible to color \\(6k\\) cells of a \\(6 \\times 6\\) grid such that, for any choice of three distinct rows \\(R_{1}, R_{2}, R_{3}\\) and three distinct columns \\(C_{1}, C_{2}, C_{3}\\), there exists an u... | math | {
"style": "rule",
"ground_truth": "k = 4"
} | {
"split": "train",
"index": 27216,
"solution": "The answer is \\(k=4\\). This can be obtained with the following construction: [grid image]. It now suffices to show that \\(k=5\\) and \\(k=6\\) are not attainable. The case \\(k=6\\) is clear. Assume for sake of contradiction that the \\(k=5\\) is attainable. Let... |
deepscaler | [
{
"role": "user",
"content": "For how many integer values of $n$ between 1 and 2000 inclusive does the decimal representation of $\\frac{n}{2940}$ terminate? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 27217,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The price of a bottle of \"Komfort\" fabric softener used to be 13.70 Ft, and half a capful was needed for 15 liters of water. The new composition of \"Komfort\" now costs 49 Ft, and 1 capful is needed for 8 liters of water. By what percentage has the price of the fabric soften... | math | {
"style": "rule",
"ground_truth": "1240"
} | {
"split": "train",
"index": 27218,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Assume that $p$, $q$, $r$, and $s$ are positive integers such that $p^3 = q^2$, $r^5 = s^4$, and $r - p = 31$. Determine $s - q$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2351"
} | {
"split": "train",
"index": 27219,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\\\(AB\\\\) is a chord passing through the focus of the parabola \\\\(y^{2} = 4\\sqrt{3}x\\\\), and the midpoint \\\\(M\\\\) of \\\\(AB\\\\) has an x-coordinate of \\\\(2\\\\), calculate the length of \\\\(AB\\\\. Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "4 + 2\\sqrt{3}"
} | {
"split": "train",
"index": 27220,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Complete the following questions: \n$(1)$ Calculate: $(\\sqrt{8}-\\sqrt{\\frac{1}{2}})\\div \\sqrt{2}$. \n$(2)$ Calculate: $2\\sqrt{3}\\times (\\sqrt{12}-3\\sqrt{75}+\\frac{1}{3}\\sqrt{108})$. \n$(3)$ Given $a=3+2\\sqrt{2}$ and $b=3-2\\sqrt{2}$, find the value of the algebra... | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 27221,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let positive integers $a$, $b$, $c$, and $d$ be randomly and independently selected with replacement from the set $\\{1, 2, 3,\\dots, 2000\\}$. Find the probability that the expression $ab+bc+cd+d+1$ is divisible by $4$. Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 27222,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a company, some pairs of people are friends (if $A$ is friends with $B$, then $B$ is friends with $A$). It turns out that among every set of 100 people in the company, the number of pairs of friends is odd. Find the largest possible number of people in such a company. Let's ... | math | {
"style": "rule",
"ground_truth": "101"
} | {
"split": "train",
"index": 27223,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two of the roots of the equation \\[ax^3+bx^2+cx+d=0\\]are $3$ and $-2.$ Given that $a \\neq 0,$ compute $\\frac{b+c}{a}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"split": "train",
"index": 27224,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $t_n = \\frac{n(n+1)}{2}$ be the $n$th triangular number. Find\n\\[\\frac{1}{t_1} + \\frac{1}{t_2} + \\frac{1}{t_3} + ... + \\frac{1}{t_{2002}}\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {4004}{2003}"
} | {
"split": "train",
"index": 27225,
"solution": "The problem statement and the solution provided seem unrelated. The problem statement asks for the sum of the reciprocals of the first 2002 triangular numbers, while the solution discusses logarithms and bases, which is irrelevant to the problem. Let's solve the or... |
deepscaler | [
{
"role": "user",
"content": "Sets $A$, $B$, and $C$, depicted in the Venn diagram, are such that the total number of elements in set $A$ is three times the total number of elements in set $B$. Their intersection has 1200 elements, and altogether, there are 4200 elements in the union of $A$, $B$, and $C$. I... | math | {
"style": "rule",
"ground_truth": "3825"
} | {
"split": "train",
"index": 27226,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify and rationalize the denominator: $$\\frac{1}{1+ \\frac{1}{\\sqrt{3}+1}}.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}-1"
} | {
"split": "train",
"index": 27227,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a regular octagon, there are two types of diagonals - one that connects alternate vertices (shorter) and another that skips two vertices between ends (longer). What is the ratio of the shorter length to the longer length? Express your answer as a common fraction in simplest ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 27228,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A banquet has invited 44 guests. There are 15 identical square tables, each of which can seat 1 person per side. By appropriately combining the square tables (to form rectangular or square tables), ensure that all guests are seated with no empty seats. What is the minimum numbe... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 27229,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\cos\\alpha = \\frac{5}{13}$ and $\\cos(\\alpha - \\beta) = \\frac{4}{5}$, with $0 < \\beta < \\alpha < \\frac{\\pi}{2}$,\n$(1)$ Find the value of $\\tan 2\\alpha$;\n$(2)$ Find the value of $\\cos\\beta$. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "\\frac{56}{65}"
} | {
"split": "train",
"index": 27230,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five cards with different numbers are given: $-5$, $-4$, $0$, $+4$, $+6$. Two cards are drawn from them. The smallest quotient obtained by dividing the numbers on these two cards is ____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\dfrac{3}{2}"
} | {
"split": "train",
"index": 27231,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integers $n$ with $1 \\le n \\le 2023$ is the product\n\\[\n \\prod_{k=0}^{n-1} \\left( \\left( 1 + e^{2 \\pi i k / n} \\right)^n + 1 \\right) \n\\]equal to zero, where $n$ needs to be an even multiple of $5$? Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "202"
} | {
"split": "train",
"index": 27232,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Design a set of stamps with the following requirements: The set consists of four stamps of different denominations, with denominations being positive integers. Moreover, for any denomination value among the consecutive integers 1, 2, ..., R, it should be possible to achieve it ... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 27233,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "You have a \\(2 \\times 3\\) grid filled with integers between 1 and 9. The numbers in each row and column are distinct. The first row sums to 23, and the columns sum to 14, 16, and 17 respectively.\n\nGiven the following grid:\n\\[\n\\begin{array}{c|c|c|c|}\n & 14 & 16 & 17... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 27234,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence, the sum of the first four terms is 26, the sum of the last four terms is 110, and the sum of all terms in the sequence is 187. Determine the total number of terms in the sequence. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 27235,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$ABCD$ is a parallelogram with $\\angle D$ obtuse. $M$ and $N$ are the feet of the perpendiculars from $D$ to $AB$ and $BC$ respectively. If $DB = DC = 50$ and $DA = 60$, find $DM + DN$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "88"
} | {
"split": "train",
"index": 27236,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\triangle ABC$, where the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it satisfies $a\\cos 2C+2c\\cos A\\cos C+a+b=0$.\n\n$(1)$ Find the size of angle $C$;\n\n$(2)$ If $b=4\\sin B$, find the maximum value of the area $S$ of $\\tria... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 27237,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ana and Bonita were born on the same date in different years, $n$ years apart. Last year Ana was $5$ times as old as Bonita. This year Ana's age is the square of Bonita's age. What is $n?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 27238,
"solution": "\nLet's denote Ana's age this year as $A$ and Bonita's age this year as $B$. Given that Ana and Bonita were born $n$ years apart, we have:\n\\[ A = B + n \\]\n\nFrom the problem, we know:\n1. Last year, Ana was $5$ times as old as Bonita:\n \\[ A - 1 = 5(B - 1) \... |
deepscaler | [
{
"role": "user",
"content": "Given the vertices of a rectangle are $A(0,0)$, $B(2,0)$, $C(2,1)$, and $D(0,1)$. A particle starts from the midpoint $P_{0}$ of $AB$ and moves in a direction forming an angle $\\theta$ with $AB$, reaching a point $P_{1}$ on $BC$. The particle then sequentially reflects to poin... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 27239,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of the expression $\\frac{20+16 \\times 20}{20 \\times 16}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17}{16}"
} | {
"split": "train",
"index": 27240,
"solution": "Evaluating, $\\frac{20+16 \\times 20}{20 \\times 16}=\\frac{20+320}{320}=\\frac{340}{320}=\\frac{17}{16}$. Alternatively, we could notice that each of the numerator and denominator is a multiple of 20, and so $\\frac{20+16 \\times 20}{20 \\times 16}=\\frac{20(1+16)... |
deepscaler | [
{
"role": "user",
"content": "Machine tools A, B, and C each independently process the same type of part. It is known that the probabilities of the parts processed by machine tools A, B, and C being first-class are 0.7, 0.6, and 0.8, respectively. The number of parts processed by machine tools B and C are e... | math | {
"style": "rule",
"ground_truth": "0.6517"
} | {
"split": "train",
"index": 27241,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A capacitor with a capacitance of $C_{1} = 20 \\mu$F is charged to a voltage $U_{1} = 20$ V. A second capacitor with a capacitance of $C_{2} = 5 \\mu$F is charged to a voltage $U_{2} = 5$ V. The capacitors are connected with opposite-charged plates. Determine the voltage that w... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 27242,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mitya is 11 years older than Shura. When Mitya was as old as Shura is now, he was twice as old as she was. How old is Mitya? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27.5"
} | {
"split": "train",
"index": 27243,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For positive integers $n$, let $c_{n}$ be the smallest positive integer for which $n^{c_{n}}-1$ is divisible by 210, if such a positive integer exists, and $c_{n}=0$ otherwise. What is $c_{1}+c_{2}+\\cdots+c_{210}$? Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "329"
} | {
"split": "train",
"index": 27244,
"solution": "In order for $c_{n} \\neq 0$, we must have $\\operatorname{gcd}(n, 210)=1$, so we need only consider such $n$. The number $n^{c_{n}}-1$ is divisible by 210 iff it is divisible by each of 2, 3, 5, and 7, and we can consider the order of $n$ modulo each modulus separ... |
deepscaler | [
{
"role": "user",
"content": "Let \\[\\mathbf{N} = \\begin{pmatrix} x & y & z \\\\ y & z & x \\\\ z & x & y \\end{pmatrix}\\] be a matrix with real entries such that $\\mathbf{N}^3 = \\mathbf{I}.$ If $xyz = -1$, find the possible values of $x^3 + y^3 + z^3.$ Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 27245,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the definite integral:\n\n$$\n\\int_{\\pi / 2}^{2 \\pi} 2^{8} \\cdot \\cos ^{8} x \\, dx\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "105\\pi"
} | {
"split": "train",
"index": 27246,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The bases \\( AB \\) and \\( CD \\) of the trapezoid \\( ABCD \\) are 41 and 24 respectively, and its diagonals are mutually perpendicular. Find the scalar (dot) product of the vectors \\( \\overrightarrow{AD} \\) and \\( \\overrightarrow{BC} \\). Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "984"
} | {
"split": "train",
"index": 27247,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many whole numbers between 1 and 1000 do not contain the digit 1? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "728"
} | {
"split": "train",
"index": 27248,
"solution": "1. **Identify the Range and Condition**: We need to find how many whole numbers between 1 and 1000 do not contain the digit 1.\n\n2. **Consider the Number of Digits**: Numbers between 1 and 1000 can have 1, 2, or 3 digits. We will consider each case separately.\n\n... |
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