data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Calculate $7 \\cdot 9\\frac{2}{5}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "65\\frac{4}{5}"
} | {
"split": "train",
"index": 38249,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of positive real numbers. Let $f : S \\to \\mathbb{R}$ be a function such that\n\\[f(x) f(y) = f(xy) + 2023 \\left( \\frac{1}{x} + \\frac{1}{y} + 2022 \\right)\\]\nfor all $x, y > 0.$\n\nLet $n$ be the number of possible values of $f(2)$, and let $s$ be the s... | math | {
"style": "rule",
"ground_truth": "\\frac{4047}{2}"
} | {
"split": "train",
"index": 38250,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two points $A, B$ are randomly chosen on a circle with radius $100.$ For a positive integer $x$ , denote $P(x)$ as the probability that the length of $AB$ is less than $x$ . Find the minimum possible integer value of $x$ such that $\\text{P}(x) > \\frac{2}{3}$ . Le... | math | {
"style": "rule",
"ground_truth": "174"
} | {
"split": "train",
"index": 38251,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In right triangle $DEF$, where $DE = 15$, $DF = 9$, and $EF = 12$ units, how far is point $F$ from the midpoint of segment $DE$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7.5"
} | {
"split": "train",
"index": 38252,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $\\frac{\\frac{1}{2}-\\frac{1}{3}}{\\frac{1}{3}-\\frac{1}{4}} \\times \\frac{\\frac{1}{4}-\\frac{1}{5}}{\\frac{1}{5}-\\frac{1}{6}} \\times \\frac{\\frac{1}{6}-\\frac{1}{7}}{\\frac{1}{7}-\\frac{1}{8}} \\times \\ldots \\times \\frac{\\frac{1}{2004}-\\frac{1}{200... | math | {
"style": "rule",
"ground_truth": "1004"
} | {
"split": "train",
"index": 38253,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the function $f(x)=x^{2}-m\\cos x+m^{2}+3m-8$ has a unique zero, then the set of real numbers $m$ that satisfy this condition is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\{2\\}"
} | {
"split": "train",
"index": 38254,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sequence $\\{a\\_n\\}$ is a monotonically increasing arithmetic sequence, the probability that any three terms taken from $\\{a\\_1, a\\_2, a\\_3, a\\_4, a\\_5, a\\_6, a\\_7\\}$ will leave the remaining four terms as a monotonically increasing arithmetic sequence... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{7}"
} | {
"split": "train",
"index": 38255,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse in the Cartesian coordinate system $xoy$, its center is at the origin, the left focus is $F(-\\sqrt{3},0)$, and the right vertex is $D(2,0)$. Let point $A(1,\\frac{1}{2})$.\n\n(1) Find the standard equation of the ellipse;\n(2) If $P$ is a moving point on the e... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 38256,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $2500$ are not factorial tails? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "499"
} | {
"split": "train",
"index": 38257,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are two opaque bags, bag A contains 2 red balls and 3 white balls, bag B contains 3 red balls and 2 white balls, and all balls are of the same size and texture. \n$(1)$ From these 10 balls, 4 balls are randomly selected. Let event A be \"exactly 2 red balls are drawn fro... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{42}"
} | {
"split": "train",
"index": 38258,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square floor is tiled with a large number of regular hexagonal tiles, which are either blue or white. Each blue tile is surrounded by 6 white tiles, and each white tile is surrounded by 3 white and 3 blue tiles. Determine the ratio of the number of blue tiles to the number of... | math | {
"style": "rule",
"ground_truth": "1: 2"
} | {
"split": "train",
"index": 38259,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In quadrilateral \\(ABCD\\), \\(AB = BC\\), \\(\\angle A = \\angle B = 20^{\\circ}\\), \\(\\angle C = 30^{\\circ}\\). The extension of side \\(AD\\) intersects \\(BC\\) at point... Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 38260,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest three-digit integer starting with 8 that is divisible by each of its distinct, non-zero digits except for 7. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "864"
} | {
"split": "train",
"index": 38261,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A deck of fifty-two cards consists of four $1$'s, four $2$'s,..., and four $13$'s. A matching pair (two cards with the same number) is removed from the deck. Determine the probability that two randomly selected cards from the remaining deck also form a pair, and express the res... | math | {
"style": "rule",
"ground_truth": "1298"
} | {
"split": "train",
"index": 38262,
"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": 38263,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The segment connecting the centers of two intersecting circles is divided by their common chord into segments equal to 5 and 2. Find the common chord, given that the radius of one circle is twice the radius of the other. Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{3}"
} | {
"split": "train",
"index": 38264,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}=(\\cos x, \\sin x)$ and $\\overrightarrow{b}=(\\sqrt{3}\\cos x, 2\\cos x-\\sqrt{3}\\sin x)$, let $f(x)=\\overrightarrow{a}\\cdot\\overrightarrow{b}$. \n$(1)$ Find the interval where $f(x)$ is monotonically decreasing. \n$(2)$ If the maximum ... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 38265,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose \\( a, b \\), and \\( c \\) are real numbers with \\( a < b < 0 < c \\). Let \\( f(x) \\) be the quadratic function \\( f(x) = (x-a)(x-c) \\) and \\( g(x) \\) be the cubic function \\( g(x) = (x-a)(x-b)(x-c) \\). Both \\( f(x) \\) and \\( g(x) \\) have the same \\( y \\... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{3}"
} | {
"split": "train",
"index": 38266,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A set consists of 120 distinct blocks. Each block is one of 3 materials (plastic, wood, metal), 3 sizes (small, medium, large), 4 colors (blue, green, red, yellow), and 5 shapes (circle, hexagon, square, triangle, rectangle). How many blocks in the set differ from the 'wood sma... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 38267,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\sqrt{2}\\cos(2x - \\frac{\\pi}{4})$, where $x \\in \\mathbb{R}$,\n1. Find the smallest positive period of the function $f(x)$ and its intervals of monotonically increasing values.\n2. Find the minimum and maximum values of the function $f(x)$ on the... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 38268,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A student needs to provide his waist size in centimeters for a customized lab coat, based on the measurements in inches. If there are $10$ inches in a foot and $25$ centimeters in a foot, then what size should the student specify in centimeters if his waist size is $40$ inches?... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 38269,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One day in the 20th century (1900-1999), a younger brother said to his older brother: \"Brother, look, if you add up the four digits of the year you were born, it gives my age.\" The elder brother responded: \"Dear brother, you are right! The same applies to me, if I add up the... | math | {
"style": "rule",
"ground_truth": "1941"
} | {
"split": "train",
"index": 38270,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define the sequence \\( b_1, b_2, b_3, \\ldots \\) by \\( b_n = \\sum\\limits_{k=1}^n \\cos{k} \\), where \\( k \\) represents radian measure. Find the index of the 50th term for which \\( b_n < 0 \\). Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "314"
} | {
"split": "train",
"index": 38271,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the remainder when the sum $1^2 + 2^2 + 3^2 + \\dots + 10^2$ is divided by 11? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 38272,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $F_1$ and $F_2$ are the left and right foci of the hyperbola $E: \\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$, and point $M$ is on $E$ with $MF_1$ perpendicular to the x-axis and $\\sin \\angle MF_2F_1 = \\frac{1}{3}$, find the eccentricity of $E$. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 38273,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of angles $\\theta$ between 0 and $2 \\pi$, other than integer multiples of $\\pi / 2$, such that the quantities $\\sin \\theta, \\cos \\theta$, and $\\tan \\theta$ form a geometric sequence in some order. Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 38274,
"solution": "If $\\sin \\theta, \\cos \\theta$, and $\\tan \\theta$ are in a geometric progression, then the product of two must equal the square of the third. Using this criterion, we have 3 cases. - Case 1: $\\sin \\theta \\cdot \\tan \\theta=\\cos ^{2} \\theta$. This implies... |
deepscaler | [
{
"role": "user",
"content": "For a given positive integer $k$ find, in terms of $k$, the minimum value of $N$ for which there is a set of $2k + 1$ distinct positive integers that has sum greater than $N$ but every subset of size $k$ has sum at most $\\tfrac{N}{2}.$ Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "2k^3 + 3k^2 + 3k"
} | {
"split": "train",
"index": 38275,
"solution": "\nGiven a positive integer \\( k \\), we need to find the minimum value of \\( N \\) such that there exists a set \\( S \\) of \\( 2k + 1 \\) distinct positive integers satisfying the following conditions:\n1. The sum of all elements in \\( S \\) is greater than \\... |
deepscaler | [
{
"role": "user",
"content": "If the function $f(x)$ satisfies $f(x) + 2f\\left(\\frac{1}{x}\\right) = 2x + 1$, find the value of $f(2)$. 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": 38276,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the polynomial $x^2-5x+t$ has only positive integer roots, find the average of all distinct possible values of $t$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 38277,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Veronica put on five rings: one on her little finger, one on her middle finger, and three on her ring finger. In how many different orders can she take them all off one by one? \n\nA) 16\nB) 20\nC) 24\nD) 30\nE) 45 Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 38278,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ that satisfies $a_n-(-1)^n a_{n-1}=n$ ($n\\geqslant 2, n\\in \\mathbb{N}^*$), and $S_n$ is the sum of the first $n$ terms of $\\{a_n\\}$, then $S_{40}=$_______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "440"
} | {
"split": "train",
"index": 38279,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When simplified, $\\log{8} \\div \\log{\\frac{1}{8}}$ becomes: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 38280,
"solution": "1. **Rewrite the fraction inside the logarithm**: We start by recognizing that $\\frac{1}{8}$ can be expressed as $8^{-1}$. Therefore, the expression $\\log{\\frac{1}{8}}$ can be rewritten using the property of logarithms that $\\log{a^{-b}} = -b \\log{a}$:\n \\[... |
deepscaler | [
{
"role": "user",
"content": "Given the function \\( f(x) = x^2 \\cos \\frac{\\pi x}{2} \\), and the sequence \\(\\left\\{a_n\\right\\}\\) in which \\( a_n = f(n) + f(n+1) \\) where \\( n \\in \\mathbf{Z}_{+} \\). Find the sum of the first 100 terms of the sequence \\(\\left\\{a_n\\right\\}\\), denoted as \... | math | {
"style": "rule",
"ground_truth": "10200"
} | {
"split": "train",
"index": 38281,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The coefficient of $x^3$ in the expansion of $(x^2-x-2)^4$ is __________ (fill in the answer with a number). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-40"
} | {
"split": "train",
"index": 38282,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Use the Horner's method to calculate the value of the polynomial $f(x) = 7x^7 + 6x^6 + 5x^5 + 4x^4 + 3x^3 + 2x^2 + x$ when $x = 3$, and find the value of $v_3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "262"
} | {
"split": "train",
"index": 38283,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{m},$ $\\mathbf{n},$ and $\\mathbf{p}$ be unit vectors such that the angle between $\\mathbf{m}$ and $\\mathbf{n}$ is $\\alpha,$ and the angle between $\\mathbf{p}$ and $\\mathbf{m} \\times \\mathbf{n}$ is also $\\alpha.$ If $\\mathbf{n} \\cdot (\\mathbf{p} \\time... | math | {
"style": "rule",
"ground_truth": "30^\\circ"
} | {
"split": "train",
"index": 38284,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum value of $S$ such that any finite number of small squares with a total area of $S$ can be placed inside a unit square $T$ with side length 1, in such a way that no two squares overlap. 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": 38285,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system \\( xOy \\), the area of the region corresponding to the set of points \\( K = \\{(x, y) \\mid (|x| + |3y| - 6)(|3x| + |y| - 6) \\leq 0 \\} \\) is ________. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 38286,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers on the faces of this cube are consecutive whole numbers. The sum of the two numbers on each of the three pairs of opposite faces are equal. The sum of the six numbers on this cube is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 38287,
"solution": "1. **Identify the properties of the cube**: The problem states that the numbers on the faces of the cube are consecutive whole numbers and the sum of the numbers on each pair of opposite faces is equal.\n\n2. **Determine possible sets of numbers**: Since the number... |
deepscaler | [
{
"role": "user",
"content": "Coordinate axes (without any marks, with the same scale) and the graph of a quadratic trinomial $y = x^2 + ax + b$ are drawn in the plane. The numbers $a$ and $b$ are not known. How to draw a unit segment using only ruler and compass? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 38288,
"solution": "\nTo construct a unit segment using only a ruler and compass, given the graph of the quadratic trinomial \\( y = x^2 + ax + b \\) without any coordinate marks, we can use geometric properties of the parabola.\n\n### Steps:\n\n1. **Identify the Vertex and Axis of Sy... |
deepscaler | [
{
"role": "user",
"content": "Let $T$ be a triangle whose vertices have integer coordinates, such that each side of $T$ contains exactly $m$ points with integer coordinates. If the area of $T$ is less than $2020$ , determine the largest possible value of $m$ .\n\n Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 38289,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $S_{n}$ is the sum of the first $n$ terms of the sequence $\\{a_{n}\\}$, and ${S}_{n}=\\frac{{n}^{2}+3n}{2}$. \n$(1)$ Find the general formula for the sequence $\\{a_{n}\\}$; \n$(2)$ Let $T_{n}$ be the sum of the first $n$ terms of the sequence $\\{\\frac{1}{{a}_{n... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 38290,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of an equation \\[\\sqrt{(x-3)^2 + (y+4)^2} + \\sqrt{(x+5)^2 + (y-8)^2} = 20.\\]is an ellipse. What is the distance between its foci? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{13}"
} | {
"split": "train",
"index": 38291,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $1-2+3-4+\\dots+100-101$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 38292,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define an ordered quadruple of integers $(a, b, c, d)$ as interesting if $1 \\le a<b<c<d \\le 10$, and $a+d>b+c$. How many interesting ordered quadruples are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 38293,
"solution": "We first start out when the value of $a=1$.\n Doing casework, we discover that $d=5,6,7,8,9,10$. We quickly find a pattern.\n Now, doing this for the rest of the values of $a$ and $d$, we see that the answer is simply:\n$(1)+(2)+(1+3)+(2+4)+(1+3+5)+(2+4+6)+(1)+(2)+... |
deepscaler | [
{
"role": "user",
"content": "The perpendicular bisectors of the sides of triangle $DEF$ meet its circumcircle at points $D'$, $E'$, and $F'$, respectively. If the perimeter of triangle $DEF$ is 42 and the radius of the circumcircle is 10, find the area of hexagon $DE'F'D'E'F$. Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "105"
} | {
"split": "train",
"index": 38294,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x,$ $y,$ $z$ be real numbers such that $4x^2 + y^2 + 16z^2 = 1.$ Find the maximum value of\n\\[7x + 2y + 8z.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"split": "train",
"index": 38295,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the following conditions:①$asinB=bsin({A+\\frac{π}{3}})$; ②$S=\\frac{{\\sqrt{3}}}{2}\\overrightarrow{BA}•\\overrightarrow{CA}$; ③$c\\tan A=\\left(2b-c\\right)\\tan C$. Choose one of the three conditions and fill in the blank below, then answer the following questions.<br/... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 38296,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Steve wrote the digits $1$, $2$, $3$, $4$, and $5$ in order repeatedly from left to right, forming a list of $10,000$ digits, beginning $123451234512\\ldots.$ He then erased every third digit from his list (that is, the $3$rd, $6$th, $9$th, $\\ldots$ digits from the left), then... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 38297,
"solution": "1. **Initial Setup and First Erasure:**\n - Steve starts with a repeating sequence of digits: $12345$.\n - The sequence repeats every $5$ digits.\n - First, every third digit is erased. To understand the pattern after this erasure, consider the least common m... |
deepscaler | [
{
"role": "user",
"content": "There are lily pads in a row numbered $0$ to $11$, in that order. There are predators on lily pads $3$ and $6$, and a morsel of food on lily pad $10$. Fiona the frog starts on pad $0$, and from any given lily pad, has a $\\frac{1}{2}$ chance to hop to the next pad, and an equal... | math | {
"style": "rule",
"ground_truth": "\\frac{15}{256}"
} | {
"split": "train",
"index": 38298,
"solution": "1. **Define the problem**: Fiona the frog starts at lily pad $0$ and can hop to the next pad or jump two pads forward with equal probability of $\\frac{1}{2}$. She must reach pad $10$ without landing on pads $3$ or $6$.\n\n2. **Calculate probability to reach pad $2... |
deepscaler | [
{
"role": "user",
"content": "Find the matrix $\\mathbf{M}$ such that\n\\[\\mathbf{M} \\mathbf{v} = \\begin{pmatrix} 5 \\\\ 2 \\\\ -8 \\end{pmatrix} \\times \\mathbf{v}\\]for all vectors $\\mathbf{v}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 0 & 8 & 2 \\\\ -8 & 0 & -5 \\\\ -2 & 5 & 0 \\end{pmatrix}"
} | {
"split": "train",
"index": 38299,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A_n$ be the sum of the first $n$ terms of the geometric series\n\\[704 + \\frac{704}{2} + \\frac{704}{4} + \\dotsb,\\]and let $B_n$ be the sum of the first $n$ terms of the geometric series\n\\[1984 - \\frac{1984}{2} + \\frac{1984}{4} - \\dotsb.\\]Compute the value of $n \... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 38300,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two circles have the same center $C.$ (Circles which have the same center are called concentric.) The larger circle has radius $10$ and the smaller circle has radius $6.$ Determine the area of the ring between these two circles. [asy]\nimport graph;\nfilldraw(circle((0,0),10), ... | math | {
"style": "rule",
"ground_truth": "64\\pi"
} | {
"split": "train",
"index": 38301,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a student speech competition held at a school, there were a total of 7 judges. The final score for a student was the average score after removing the highest and the lowest scores. The scores received by a student were 9.6, 9.4, 9.6, 9.7, 9.7, 9.5, 9.6. The mode of this data... | math | {
"style": "rule",
"ground_truth": "9.6"
} | {
"split": "train",
"index": 38302,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $4 \\times 4$ window is made out of 16 square windowpanes. How many ways are there to stain each of the windowpanes, red, pink, or magenta, such that each windowpane is the same color as exactly two of its neighbors? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 38303,
"solution": "For the purpose of explaining this solution, let's label the squares as 11121314 21222324 31323334 41424344. Note that since the corner squares $11,14,41,44$ each only have two neighbors, each corner square is the same color as both of its neighbors. This corner sq... |
deepscaler | [
{
"role": "user",
"content": "A cube with an edge length of 2 decimeters is first cut 4 times horizontally, then 5 times vertically. The total surface area of all the small pieces after cutting is ____ square decimeters. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 38304,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Esmeralda has created a special knight to play on quadrilateral boards that are identical to chessboards. If a knight is in a square then it can move to another square by moving 1 square in one direction and 3 squares in a perpendicular direction (which is a diagonal of a $2\\t... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 38305,
"solution": "\nTo solve this problem, we will analyze the movement of Esmeralda's special knight on a \\(5 \\times 5\\) board and determine the longest possible path without revisiting any square. The knight moves in an \"L\" shape spanning across a \\(2 \\times 4\\) rectangle,... |
deepscaler | [
{
"role": "user",
"content": "Arrange the natural numbers according to the following triangular pattern. What is the sum of the numbers in the 10th row?\n\n```\n 1\n 2 3 4\n 5 6 7 8 9\n 10 11 12 13 14 15 16\n17 18 19 20 21 22 23 24 25\n...\n``` Let's th... | math | {
"style": "rule",
"ground_truth": "1729"
} | {
"split": "train",
"index": 38306,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Person A can only be in the first or second position, and person B can only be in the second or third position. Find the total number of different possible arrangements of five people in a row. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 38307,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the distribution list of the random variable $X$, $P(X=\\frac{k}{5})=ak$, where $k=1,2,3,4,5$.\n\n1. Find the value of the constant $a$.\n2. Find $P(X\\geqslant\\frac{3}{5})$.\n3. Find $P(\\frac{1}{10}<X<\\frac{7}{10})$. Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"split": "train",
"index": 38308,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a unique positive real number $x$ such that the three numbers $\\log_8{2x}$, $\\log_4{x}$, and $\\log_2{x}$, in that order, form a geometric progression with positive common ratio. The number $x$ can be written as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime ... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 38309,
"solution": "If we set $x=2^y$, we can obtain three terms of a geometric sequence through logarithm properties. The three terms are \\[\\frac{y+1}{3}, \\frac{y}{2}, y.\\] In a three-term geometric sequence, the middle term squared is equal to the product of the other two terms,... |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{a} = \\begin{pmatrix} 3 \\\\ p \\\\ -1 \\end{pmatrix}$ and $\\mathbf{b} = \\begin{pmatrix} 2 \\\\ 1 \\\\ q \\end{pmatrix}$ be vectors that are equal in magnitude, and orthogonal. Enter the ordered pair $(p,q).$ Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "\\left( -\\frac{31}{12}, \\frac{41}{12} \\right)"
} | {
"split": "train",
"index": 38310,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $M = 1! \\times 2! \\times 3! \\times 4! \\times 5! \\times 6! \\times 7! \\times 8! \\times 9!$, calculate the number of divisors of $M$ that are perfect squares. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "672"
} | {
"split": "train",
"index": 38311,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2010,0),(2010,2011),$ and $(0,2011)$. What is the probability that $x > 3y$? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "\\frac{335}{2011}"
} | {
"split": "train",
"index": 38312,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all functions $f:\\mathbb{N}\\rightarrow \\mathbb{N}$ such that the inequality $$f(x)+yf(f(x))\\le x(1+f(y))$$\nholds for all positive integers $x, y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = x"
} | {
"split": "train",
"index": 38313,
"solution": "\nLet's analyze the problem by working with the given inequality:\n\n\\[\nf(x) + y f(f(x)) \\le x(1 + f(y))\n\\]\n\nfor all positive integers \\(x, y\\).\n\nTo find all functions \\(f: \\mathbb{N} \\rightarrow \\mathbb{N}\\) satisfying this inequality, we will firs... |
deepscaler | [
{
"role": "user",
"content": "For $-1<r<1$, let $S(r)$ denote the sum of the geometric series \\[12+12r+12r^2+12r^3+\\cdots .\\]Let $a$ between $-1$ and $1$ satisfy $S(a)S(-a)=2016$. Find $S(a)+S(-a)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "336"
} | {
"split": "train",
"index": 38314,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A \\(3\\times 5\\) rectangle and a \\(4\\times 6\\) rectangle need to be contained within a square without any overlapping at their interior points, and the square's sides are parallel to the sides of the given rectangles. Determine the smallest possible area of this square. Le... | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 38315,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\sum_{n = 1}^\\infty \\frac{1}{n(n + 2)}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 38316,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that spinner A contains the numbers 4, 5, 6, spinner B contains the numbers 1, 2, 3, 4, 5, and spinner C can take numbers from the set 7, 8, 9 if spinner B lands on an odd number and the set {6, 8} if spinner B lands on an even number, find the probability that the sum of... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{15}"
} | {
"split": "train",
"index": 38317,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the following arrays, each composed of three numbers: $(1,2,3)$, $(2,4,6)$, $(3,8,11)$, $(4,16,20)$, $(5,32,37)$, ..., $(a\\_n,b\\_n,c\\_n)$. If the sum of the first $n$ terms of the sequence $\\{c\\_n\\}$ is denoted as $M\\_n$, find the value of $M\\_{10}$. Let's think s... | math | {
"style": "rule",
"ground_truth": "2101"
} | {
"split": "train",
"index": 38318,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider an unusual biased coin, with probability $p$ of landing heads, probability $q \\leq p$ of landing tails, and probability \\frac{1}{6}$ of landing on its side (i.e. on neither face). It is known that if this coin is flipped twice, the likelihood that both flips will hav... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 38319,
"solution": "The probability that both flips are the same is $p^{2}+q^{2}+\\frac{1}{36}$. For this to be \\frac{1}{2}$, we must have $$p^{2}+q^{2}+\\frac{1}{36}=p^{2}+\\left(\\frac{5}{6}-p\\right)^{2}+\\frac{1}{36}=\\frac{1}{2}$$ Using the quadratic formula, $p=\\frac{2}{3}$ or... |
deepscaler | [
{
"role": "user",
"content": "Let $A M O L$ be a quadrilateral with $A M=10, M O=11$, and $O L=12$. Given that the perpendicular bisectors of sides $A M$ and $O L$ intersect at the midpoint of segment $A O$, find the length of side LA. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "\\sqrt{77}"
} | {
"split": "train",
"index": 38320,
"solution": "Let $D$ be the midpoint of $A M$ and $E$ be the midpoint of $A O$. Then, we note that $A D E \\sim A M O$, so $M$ is a right angle. Similarly, $L$ is a right angle. Consequently, we get that $$A O^{2}=O M^{2}+A M^{2} \\Rightarrow A L=\\sqrt{A O^{2}-O L^{2}}=\\sqrt{... |
deepscaler | [
{
"role": "user",
"content": "Let $a$ be a real number. Find the minimum value of $\\int_0^1 |ax-x^3|dx$ .\n\nHow many solutions (including University Mathematics )are there for the problem?\n\nAny advice would be appreciated. :) Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "1/8"
} | {
"split": "train",
"index": 38321,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the definite integral:\n\n$$\n\\int_{0}^{2 \\pi} \\sin^{6} x \\cos^{2} x \\, dx\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{64}"
} | {
"split": "train",
"index": 38322,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The increasing sequence \\( T = 2, 3, 5, 6, 7, 8, 10, 11, \\ldots \\) consists of all positive integers which are not perfect squares. What is the 2012th term of \\( T \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2057"
} | {
"split": "train",
"index": 38323,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that 2 students exercised 0 days, 4 students exercised 1 day, 5 students exercised 2 days, 3 students exercised 4 days, 7 students exercised 5 days, and 2 students exercised 6 days, calculate the average number of days exercised last week by the students in Ms. Brown's cl... | math | {
"style": "rule",
"ground_truth": "3.17"
} | {
"split": "train",
"index": 38324,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the merchant purchased $1200$ keychains at $0.15$ each and desired to reach a target profit of $180$, determine the minimum number of keychains the merchant must sell if each is sold for $0.45$. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "800"
} | {
"split": "train",
"index": 38325,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers $n$ satisfy the condition $100 < n < 200$ and the condition $n$ has the same remainder whether it is divided by $6$ or by $8$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 38326,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the picture, there is a grid consisting of 25 small equilateral triangles. How many rhombuses can be made 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": 38327,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $α+β= \\frac {π}{3}$ and $tanα+tanβ=2$, find the value of $cos(α-β)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {3}-1}{2}"
} | {
"split": "train",
"index": 38328,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The expression \\(\\frac{3}{10}+\\frac{3}{100}+\\frac{3}{1000}\\) can be simplified by converting each fraction to a decimal, and then calculating the sum. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.333"
} | {
"split": "train",
"index": 38329,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The altitude to the hypotenuse of a triangle with angles of 30 and 60 degrees is 3 units. What is the area of the triangle, in square units? Express your answer in simplest radical form. [asy]\nunitsize(6mm);\ndefaultpen(linewidth(.7pt)+fontsize(8pt));\n\nreal r=2*sqrt(3);\n\np... | math | {
"style": "rule",
"ground_truth": "6\\sqrt{3}"
} | {
"split": "train",
"index": 38330,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\{a_n\\}$ is an arithmetic sequence, with the first term $a_1 > 0$, $a_{2012} + a_{2013} > 0$, and $a_{2012} \\cdot a_{2013} < 0$, then the largest natural number $n$ for which the sum of the first $n$ terms $S_n > 0$ is. Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "2012"
} | {
"split": "train",
"index": 38331,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\Delta ABC$, $AC = BC$, $m\\angle DCB = 40^{\\circ}$, and $CD \\parallel AB$. What is the number of degrees in $m\\angle ECD$?\n\n[asy] pair A,B,C,D,E; B = dir(-40); A = dir(-140); D = (.5,0); E = .4 * dir(40);\ndraw(C--B--A--E,EndArrow); draw(C--D,EndArrow);\nlabel(\"$A$\... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 38332,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the approximate ratio of the three cases drawn, DD, Dd, dd, is 1:2:1, calculate the probability of drawing dd when two students who have drawn cards are selected, and one card is drawn from each of these two students. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 38333,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the fraction that equals $0.\\overline{73}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{73}{99}"
} | {
"split": "train",
"index": 38334,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the general term formula of the sequence $\\{a\\_n\\}$, where $a\\_n=n\\cos \\frac {nπ}{2}$, and the sum of the first $n$ terms is represented by $S\\_n$, find the value of $S\\_{2016}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1008"
} | {
"split": "train",
"index": 38335,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $x,$ $y,$ and $z$ are three positive numbers that satisfy the equations $xyz = 1,$ $x + \\frac {1}{z} = 5,$ and $y + \\frac {1}{x} = 29.$ Then $z + \\frac {1}{y} = \\frac {m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.\n note: this ... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 38336,
"solution": "We can rewrite $xyz=1$ as $\\frac{1}{z}=xy$.\n Substituting into one of the given equations, we have \\[x+xy=5\\] \\[x(1+y)=5\\] \\[\\frac{1}{x}=\\frac{1+y}{5}.\\]\n We can substitute back into $y+\\frac{1}{x}=29$ to obtain \\[y+\\frac{1+y}{5}=29\\] \\[5y+1+y=145\\... |
deepscaler | [
{
"role": "user",
"content": "Determine the area, in square units, of triangle $PQR$, where the coordinates of the vertices are $P(-3, 4)$, $Q(4, 9)$, and $R(5, -3)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "44.5"
} | {
"split": "train",
"index": 38337,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A box contains $28$ red balls, $20$ green balls, $19$ yellow balls, $13$ blue balls, $11$ white balls, and $9$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $15$ balls of a single color will be dr... | math | {
"style": "rule",
"ground_truth": "76"
} | {
"split": "train",
"index": 38338,
"solution": "To solve this problem, we need to determine the minimum number of balls that must be drawn to ensure that at least 15 balls of one color are drawn. We will consider the worst-case scenario where we draw the maximum number of balls possible without reaching 15 balls... |
deepscaler | [
{
"role": "user",
"content": "What three-digit number with units digit 4 and hundreds digit 5 is divisible by 8 and has an even tens digit? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "544"
} | {
"split": "train",
"index": 38339,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consecutive powers of 3 are added to form this sequence: $3^0,3^0+ 3^1, 3^0+ 3^1+ 3^2$, and so on. What is the simplified value of the fourth term of the sequence? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 38340,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an isosceles trapezoid, the lengths of the bases are 9 cm and 21 cm, and the height is 8 cm. Find the radius of the circumscribed circle around the trapezoid. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10.625"
} | {
"split": "train",
"index": 38341,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ with the first term $\\frac{3}{2}$ and common ratio $-\\frac{1}{2}$, and the sum of the first $n$ terms is $S_n$, then when $n\\in N^*$, the sum of the maximum and minimum values of $S_n - \\frac{1}{S_n}$ is ______. Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 38342,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\\[\\frac{\\tan 96^{\\circ} - \\tan 12^{\\circ} \\left( 1 + \\frac{1}{\\sin 6^{\\circ}} \\right)}{1 + \\tan 96^{\\circ} \\tan 12^{\\circ} \\left( 1 + \\frac{1}{\\sin 6^{\\circ}} \\right)} =\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 38343,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five consecutive two-digit positive integers, each less than 30, are not prime. What is the largest of these five integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 38344,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of $a, b$ and $c$ is equal to a number from the list $3^{1}, 3^{2}, 3^{3}, 3^{4}, 3^{5}, 3^{6}, 3^{7}, 3^{8}$. There are $N$ triples $(a, b, c)$ with $a \\leq b \\leq c$ for which each of $\\frac{ab}{c}, \\frac{ac}{b}$ and $\\frac{bc}{a}$ is equal to an integer. What is th... | math | {
"style": "rule",
"ground_truth": "86"
} | {
"split": "train",
"index": 38345,
"solution": "We write $a=3^{r}, b=3^{s}$ and $c=3^{t}$ where each of $r, s, t$ is between 1 and 8, inclusive. Since $a \\leq b \\leq c$, then $r \\leq s \\leq t$. Next, we note that $\\frac{ab}{c}=\\frac{3^{r} 3^{s}}{3^{t}}=3^{r+s-t}$, $\\frac{ac}{b}=\\frac{3^{r} 3^{t}}{3^{s}}=... |
deepscaler | [
{
"role": "user",
"content": "The base nine numbers $125_9$ and $33_9$ need to be multiplied and the result expressed in base nine. What is the base nine sum of the digits of their product? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 38346,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the value of $$1 \\cdot 2-2 \\cdot 3+3 \\cdot 4-4 \\cdot 5+\\cdots+2001 \\cdot 2002$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2004002"
} | {
"split": "train",
"index": 38347,
"solution": "2004002. Rewrite the expression as $$2+3 \\cdot(4-2)+5 \\cdot(6-4)+\\cdots+2001 \\cdot(2002-2000)$$ $$=2+6+10+\\cdots+4002$$ This is an arithmetic progression with $(4002-2) / 4+1=1001$ terms and average 2002, so its sum is $1001 \\cdot 2002=2004002$."
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ $c,$ and $d$ be the roots of \\[x^4 + 8x^3 + 9x^2 + 5x + 4 = 0.\\]Find the value of \\[\\frac{1}{ab} + \\frac{1}{ac} + \\frac{1}{ad} + \\frac{1}{bc} + \\frac{1}{bd} + \\frac{1}{cd}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\tfrac 94"
} | {
"split": "train",
"index": 38348,
"solution": ""
} |
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