data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\\cos A= \\frac{4}{5}$.\n\n(1) Find the value of $\\sin ^{2} \\frac{B+C}{2}+\\cos 2A$;\n\n(2) If $b=2$, the area of $\\triangle ABC$ is $S=3$, find $a$. Let's think step... | math | {
"style": "rule",
"ground_truth": "\\sqrt{13}"
} | {
"split": "train",
"index": 39149,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( \\triangle ABC \\), point \\( E \\) is on side \\( AB \\) with \\( AE = 1 \\) and \\( EB = 2 \\). Suppose points \\( D \\) and \\( F \\) are on sides \\( AC \\) and \\( BC \\) respectively, and \\( DE \\parallel BC \\) and \\( EF \\parallel AC \\). What is the r... | math | {
"style": "rule",
"ground_truth": "4: 9"
} | {
"split": "train",
"index": 39150,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiao Ming invests in selling a type of eye-protecting desk lamp with a cost price of $20$ yuan per unit. During the sales process, he found that the relationship between the monthly sales volume $y$ (units) and the selling price $x$ (yuan) can be approximated as a linear functi... | math | {
"style": "rule",
"ground_truth": "2160"
} | {
"split": "train",
"index": 39151,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many triangles with positive area have all their vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $5$, inclusive?\n$\\textbf{(A)}\\ 2128 \\qquad\\textbf{(B)}\\ 2148 \\qquad\\textbf{(C)}\\ 2160 \\qquad\\textbf{(D)}\\ 2200 \\q... | math | {
"style": "rule",
"ground_truth": "2148"
} | {
"split": "train",
"index": 39152,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the set of four-digit numbers composed of the digits 0, 1, 2, 3, 4, 5 without any repetition, there are a total of numbers that are not divisible by 5. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "192"
} | {
"split": "train",
"index": 39153,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{\\alpha}$, $\\overrightarrow{\\beta}$, $\\overrightarrow{\\gamma}$ satisfy $|\\overrightarrow{\\alpha}|=1$, $|\\overrightarrow{\\alpha}-\\overrightarrow{\\beta}|=|\\overrightarrow{\\beta}|$, $(\\overrightarrow{\\alpha}-\\overrightarrow{\\gamma}) ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 39154,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the line that passes through the points $(2,7)$ and $(a, 3a)$ has a slope of 2, what is the value of $a$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 39155,
"solution": "Since the slope of the line through points $(2,7)$ and $(a, 3a)$ is 2, then $\frac{3a-7}{a-2}=2$. From this, $3a-7=2(a-2)$ and so $3a-7=2a-4$ which gives $a=3$."
} |
deepscaler | [
{
"role": "user",
"content": "A sphere is inscribed in the tetrahedron whose vertices are $A = (6,0,0), B = (0,4,0), C = (0,0,2),$ and $D = (0,0,0).$ The radius of the sphere is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$ Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 39156,
"solution": "Note that the center is obviously at some point $(r,r,r)$ since it must be equidistant from the x-y, x-z, and y-z planes. Now note that the point-plane distance formula is $\\widehat{n}\\cdot \\vec{d}$ (the direction vector of the normal to the plane dotted with th... |
deepscaler | [
{
"role": "user",
"content": "Two positive integers \\( x \\) and \\( y \\) are such that:\n\n\\[ \\frac{2010}{2011} < \\frac{x}{y} < \\frac{2011}{2012} \\]\n\nFind the smallest possible value for the sum \\( x + y \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8044"
} | {
"split": "train",
"index": 39157,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $M$ be the greatest integer multiple of 9, no two of whose digits are the same. What is the remainder when $M$ is divided by 1000? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "963"
} | {
"split": "train",
"index": 39158,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find one third of 7.2, expressed as a simplified fraction or a mixed number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\frac{2}{5}"
} | {
"split": "train",
"index": 39159,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An educational center received an object with a volume of around 150 monoliths (a container designed for 150 monoliths that was almost full). Each monolith is identified either as sandy loam or loam and classified by its genesis as either marine or lake-glacial deposits. The re... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 39160,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum value of\n\\[\\cos \\theta_1 \\sin \\theta_2 + \\cos \\theta_2 \\sin \\theta_3 + \\cos \\theta_3 \\sin \\theta_4 + \\cos \\theta_4 \\sin \\theta_5 + \\cos \\theta_5 \\sin \\theta_1,\\]over all real numbers $\\theta_1,$ $\\theta_2,$ $\\theta_3,$ $\\theta_4,$ and... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"split": "train",
"index": 39161,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(a) The natural number \\( n \\) is less than 150. What is the largest remainder that the number 269 can give when divided by \\( n \\)?\n\n(b) The natural number \\( n \\) is less than 110. What is the largest remainder that the number 269 can give when divided by \\( n \\)? L... | math | {
"style": "rule",
"ground_truth": "109"
} | {
"split": "train",
"index": 39162,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We will call a ticket with a number from 000000 to 999999 excellent if the difference between some two adjacent digits of its number is 5.\n\nFind the number of excellent tickets. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "409510"
} | {
"split": "train",
"index": 39163,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( M \\) be a set of \\( n \\) points in the plane such that:\n1. There are 7 points in \\( M \\) that form the vertices of a convex heptagon.\n2. For any 5 points in \\( M \\), if these 5 points form a convex pentagon, then the interior of this convex pentagon contains at... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 39164,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Allen and Yang want to share the numbers \\(1,2,3,4,5,6,7,8,9,10\\). How many ways are there to split all ten numbers among Allen and Yang so that each person gets at least one number, and either Allen's numbers or Yang's numbers sum to an even number? Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "1022"
} | {
"split": "train",
"index": 39165,
"solution": "Since the sum of all of the numbers is odd, exactly one of Allen's sum and Yang's sum must be odd. Therefore any way of splitting the numbers up where each person receives at least one number is valid, so the answer is \\(2^{10}-2=1022\\)."
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( z_{1} \\) and \\( z_{2} \\) be complex numbers such that \\( \\left|z_{1}\\right|=3 \\), \\( \\left|z_{2}\\right|=5 \\), and \\( \\left|z_{1} + z_{2}\\right|=7 \\). Find the value of \\( \\arg \\left(\\left( \\frac{z_{2}}{z_{1}} \\right)^{3}\\right) \\). Let's think ste... | math | {
"style": "rule",
"ground_truth": "\\pi"
} | {
"split": "train",
"index": 39166,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)=x^2+3x+2$ and let $S$ be the set of integers $\\{0, 1, 2, \\dots , 25 \\}$. \nThe number of members $s$ of $S$ such that $f(s)$ has remainder zero when divided by $6$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 39167,
"solution": "1. **Understanding the Polynomial Function**: \n Given the polynomial function $f(x) = x^2 + 3x + 2$. We need to find the number of integers $s$ in the set $S = \\{0, 1, 2, \\dots, 25\\}$ such that $f(s)$ is divisible by $6$.\n\n2. **Periodicity of Polynomial Fun... |
deepscaler | [
{
"role": "user",
"content": "A circle with a radius of 3 units has its center at $(0, 0)$. Another circle with a radius of 5 units has its center at $(12, 0)$. Find the x-coordinate of the point on the $x$-axis where a line, tangent to both circles, intersects. The line should intersect the x-axis to the r... | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"split": "train",
"index": 39168,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At Euclid High School, the number of students taking the AMC 10 was $60$ in 2002, $66$ in 2003, $70$ in 2004, $76$ in 2005, $78$ in 2006, and is $85$ in 2007. Between what two consecutive years was there the largest percentage increase? Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "2002 and 2003"
} | {
"split": "train",
"index": 39169,
"solution": "To find the largest percentage increase in the number of students taking the AMC 10 between consecutive years, we calculate the percentage increase for each interval:\n\n1. **From 2002 to 2003:**\n \\[\n \\frac{66 - 60}{60} \\times 100\\% = \\frac{6}{60} \\time... |
deepscaler | [
{
"role": "user",
"content": "The line $10x + 8y = 80$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?\nA) $\\frac{18\\sqrt{41} + 40}{\\sqrt{41}}$\nB) $\\frac{360}{17}$\nC) $\\frac{107}{5}$\nD) $\\frac{43}{2}$\nE) $\\frac{281}{13}$ Let's think ste... | math | {
"style": "rule",
"ground_truth": "\\frac{18\\sqrt{41} + 40}{\\sqrt{41}}"
} | {
"split": "train",
"index": 39170,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are red and blue socks in a drawer, with a total number not exceeding 2017. If two socks are randomly drawn, the probability that they are of the same color is $\\frac{1}{2}$. What is the maximum number of red socks in the drawer? Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "990"
} | {
"split": "train",
"index": 39171,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A deck of 100 cards is labeled $1,2, \\ldots, 100$ from top to bottom. The top two cards are drawn; one of them is discarded at random, and the other is inserted back at the bottom of the deck. This process is repeated until only one card remains in the deck. Compute the expect... | math | {
"style": "rule",
"ground_truth": "\\frac{467}{8}"
} | {
"split": "train",
"index": 39172,
"solution": "Note that we can just take averages: every time you draw one of two cards, the EV of the resulting card is the average of the EVs of the two cards. This average must be of the form $$2^{\\bullet} \\cdot 1+2^{\\bullet} \\cdot 2+2^{\\bullet} \\cdot 3+\\cdots+2^{\\bul... |
deepscaler | [
{
"role": "user",
"content": "Suppose that $\\frac{2}{3}$ of $10$ bananas are worth as much as $8$ oranges. How many oranges are worth as much as $\\frac{1}{2}$ of $5$ bananas? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 39173,
"solution": "1. **Establish the given relationship**: We are given that $\\frac{2}{3}$ of $10$ bananas are worth as much as $8$ oranges. This can be written as:\n \\[\n \\frac{2}{3} \\times 10 \\text{ bananas} = 8 \\text{ oranges}\n \\]\n Simplifying the left side, we g... |
deepscaler | [
{
"role": "user",
"content": "A right rectangular prism whose surface area and volume are numerically equal has edge lengths $\\log_{2}x, \\log_{3}x,$ and $\\log_{4}x.$ What is $x?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "576"
} | {
"split": "train",
"index": 39174,
"solution": "1. **Identify the expressions for surface area and volume:**\n - The surface area (SA) of a right rectangular prism is given by:\n \\[\n SA = 2(ab + bc + ca)\n \\]\n where \\(a\\), \\(b\\), and \\(c\\) are the lengths of the edges. For this proble... |
deepscaler | [
{
"role": "user",
"content": "The distance between locations A and B is 135 kilometers. Two cars, a large one and a small one, travel from A to B. The large car departs 4 hours earlier than the small car, but the small car arrives 30 minutes earlier than the large car. The speed ratio of the small car to th... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 39175,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangle has dimensions $4$ and $2\\sqrt{3}$. Two equilateral triangles are contained within this rectangle, each with one side coinciding with the longer side of the rectangle. The triangles intersect, forming another polygon. What is the area of this polygon?\nA) $2\\sqrt{... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"split": "train",
"index": 39176,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many 5 digit positive integers are there such that each of its digits, except for the last one, is greater than or equal to the next digit? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "715"
} | {
"split": "train",
"index": 39177,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest integer whose square is 78 more than three times the integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 39178,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer \\( k \\) such that \\( (k-10)^{5026} \\geq 2013^{2013} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "55"
} | {
"split": "train",
"index": 39179,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_1 = 1$ and $a_{n+1} = a_n \\cdot p_n$ for $n \\geq 1$ where $p_n$ is the $n$ th prime number, starting with $p_1 = 2$ . Let $\\tau(x)$ be equal to the number of divisors of $x$ . Find the remainder when $$ \\sum_{n=1}^{2020} \\sum_{d \\mid a_n} \\tau (d) $$... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 39180,
"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 $5\\times 5$ square array of dots? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 39181,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least positive integer with exactly five distinct positive factors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 39182,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A dark drawer contains $90$ red socks, $70$ green socks, $50$ blue socks, and $30$ purple socks. Someone randomly selects socks without seeing their color. What is the smallest number of socks that they must pick to guarantee at least $12$ pairs? A pair of socks consists of tw... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 39183,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an isosceles triangle, one of the angles opposite an equal side is $40^{\\circ}$. How many degrees are in the measure of the triangle's largest angle? [asy] draw((0,0)--(6,0)--(3,2)--(0,0)); label(\"$\\backslash$\",(1.5,1)); label(\"{/}\",(4.5,1));\nlabel(\"$40^{\\circ}$\",(... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 39184,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many $y$-intercepts does the graph of the parabola $x = 2y^2 - 3y + 7$ have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 39185,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C$ be an acute isosceles triangle with orthocenter $H$. Let $M$ and $N$ be the midpoints of sides $\\overline{A B}$ and $\\overline{A C}$, respectively. The circumcircle of triangle $M H N$ intersects line $B C$ at two points $X$ and $Y$. Given $X Y=A B=A C=2$, compute... | math | {
"style": "rule",
"ground_truth": "2(\\sqrt{17}-1)"
} | {
"split": "train",
"index": 39186,
"solution": "Let $D$ be the foot from $A$ to $B C$, also the midpoint of $B C$. Note that $D X=D Y=M A=M B=M D=N A=N C=N D=1$. Thus, $M N X Y$ is cyclic with circumcenter $D$ and circumradius 1. $H$ lies on this circle too, hence $D H=1$. If we let $D B=D C=x$, then since $\\tr... |
deepscaler | [
{
"role": "user",
"content": "Given quadrilateral $\\Box FRDS$ with $\\triangle FDR$ being a right-angled triangle at point $D$, with side lengths $FD = 3$ inches, $DR = 4$ inches, $FR = 5$ inches, and $FS = 8$ inches, and $\\angle RFS = \\angle FDR$, find the length of RS. Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{89}"
} | {
"split": "train",
"index": 39187,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two boys and three girls stand in a row for a photo. If boy A does not stand at either end, and exactly two of the three girls are adjacent, determine the number of different arrangements. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 39188,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfy $|\\overrightarrow{a}|=3$, $|\\overrightarrow{b}|=2\\sqrt{3}$, and $\\overrightarrow{a}\\perp(\\overrightarrow{a}+\\overrightarrow{b})$, find the projection of $\\overrightarrow{b}$ in the direction of $\\ove... | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 39189,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sphere is inside a cube with an edge length of $3$, and it touches all $12$ edges of the cube. Find the volume of the sphere. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9\\sqrt{2}\\pi"
} | {
"split": "train",
"index": 39190,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p$ and $q$ satisfy $pq=9$ and $p+q=6$. What is the value of $p^2 + q^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 39191,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $x$ and $y$ are positive numbers with $xy=\\frac{1}{9}$, $x(y+1)=\\frac{7}{9}$, and $y(x+1)=\\frac{5}{18}$. What is the value of $(x+1)(y+1)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{35}{18}"
} | {
"split": "train",
"index": 39192,
"solution": "If we multiply the second and third equations together, we obtain $x(y+1)y(y+1)=\\frac{7}{9} \\cdot \\frac{5}{18}$ or $xy(x+1)(y+1)=\\frac{35}{162}$. From the first equation, $xy=\\frac{1}{9}$. Therefore, $\\frac{1}{9}(x+1)(y+1)=\\frac{35}{162}$ or $(x+1)(y+1)=9\\l... |
deepscaler | [
{
"role": "user",
"content": "In the diagram, $\\triangle PQR$ is isosceles with $PQ = PR = 39$ and $\\triangle SQR$ is equilateral with side length 30. The area of $\\triangle PQS$ is closest to: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 39193,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In trapezoid \\(ABCD\\), \\(AB\\) is parallel to \\(DC\\) and \\(\\angle DAF = 90^\\circ\\). Point \\(E\\) is on \\(DC\\) such that \\(EB = BC = CE\\). Point \\(F\\) is on \\(AB\\) such that \\(DF\\) is parallel to \\(EB\\). In degrees, what is the measure of \\(\\angle FDA\\)?... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 39194,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In right triangle $PQR$, where $PQ=8$, $QR=15$, and $\\angle Q = 90^\\circ$. Points $M$ and $N$ are midpoints of $\\overline{PQ}$ and $\\overline{PR}$ respectively; $\\overline{QN}$ and $\\overline{MR}$ intersect at point $Z$. Compute the ratio of the area of triangle $PZN$ to ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 39195,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of lattice points inside the circle $x^{2}+y^{2}=11$. Let $M$ be the greatest area of any triangle with vertices in $S$. How many triangles with vertices in $S$ have area $M$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 39196,
"solution": "The boundary of the convex hull of $S$ consists of points with $(x, y)$ or $(y, x)=(0, \\pm 3)$, $( \\pm 1, \\pm 3)$, and $( \\pm 2, \\pm 2)$. For any triangle $T$ with vertices in $S$, we can increase its area by moving a vertex not on the boundary to some point o... |
deepscaler | [
{
"role": "user",
"content": "Read the material first, then answer the question. \n$(1)$ Xiao Zhang encountered a problem when simplifying a quadratic radical: simplify $\\sqrt{5-2\\sqrt{6}}$. \nAfter thinking about it, Xiao Zhang's process of solving this problem is as follows: \n$\\sqrt{5-2\\sqrt{6}}=\... | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}+\\sqrt{2}"
} | {
"split": "train",
"index": 39197,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the sequence of six real numbers 60, 10, 100, 150, 30, and $x$ . The average (arithmetic mean) of this sequence is equal to the median of the sequence. What is the sum of all the possible values of $x$ ? (The median of a sequence of six real numbers is the average... | math | {
"style": "rule",
"ground_truth": "135"
} | {
"split": "train",
"index": 39198,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\cos 36^\\circ - \\cos 72^\\circ.$ 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": 39199,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to $\\angle A$, $\\angle B$, and $\\angle C$ are $a$, $b$, and $c$ respectively, and\n$$\na=5, \\quad b=4, \\quad \\cos(A-B)=\\frac{31}{32}.\n$$\n\nFind the area of $\\triangle ABC$. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "\\frac{15 \\sqrt{7}}{4}"
} | {
"split": "train",
"index": 39200,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$\\left(\\frac{(x+1)^{2}(x^{2}-x+1)^{2}}{(x^{3}+1)^{2}}\\right)^{2}\\cdot\\left(\\frac{(x-1)^{2}(x^{2}+x+1)^{2}}{(x^{3}-1)^{2}}\\right)^{2}$ equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 39201,
"solution": "1. **Simplify the given expression**: \n Start by simplifying the expression inside the parentheses:\n \\[\n \\left(\\frac{(x+1)^2(x^2-x+1)^2}{(x^3+1)^2}\\right)^2 \\cdot \\left(\\frac{(x-1)^2(x^2+x+1)^2}{(x^3-1)^2}\\right)^2\n \\]\n We can simplify each ... |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer that satisfies the congruence $5x \\equiv 17 \\pmod{31}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 39202,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all positive integers $n>1$ for which $\\frac{n^{2}+7 n+136}{n-1}$ is the square of a positive integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5, 37"
} | {
"split": "train",
"index": 39203,
"solution": "Write $\\frac{n^{2}+7 n+136}{n-1}=n+\\frac{8 n+136}{n-1}=n+8+\\frac{144}{n-1}=9+(n-1)+\\frac{144}{(n-1)}$. We seek to find $p$ and $q$ such that $p q=144$ and $p+q+9=k^{2}$. The possibilities are seen to be $1+144+9=154,2+72+9=83,3+48+9=60,4+36+9=49,6+24+9=39$, $8+... |
deepscaler | [
{
"role": "user",
"content": "If three standard, six-faced dice are rolled, what is the probability that the sum of the three numbers rolled is 9? 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{25}{216}"
} | {
"split": "train",
"index": 39204,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangular prism \\( P-ABC \\), \\( PA \\perp \\) plane \\( ABC \\), and \\( AC \\perp BC \\). Given \\( AC = 2 \\), the dihedral angle \\( P-BC-A \\) is \\( 60^\\circ \\), and the volume of the triangular prism \\( P-ABC \\) is \\( \\frac{4\\sqrt{6}}{3} \\). Find the sine v... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 39205,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the width of a rectangle is increased by 3 cm and the height is decreased by 3 cm, its area does not change. What would happen to the area if, instead, the width of the original rectangle is decreased by 4 cm and the height is increased by 4 cm? Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 39206,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Buses. At the stop near Absent-Minded Scientist's house, buses of two routes stop: No. 152 and No. 251. Both go to the metro station. The interval between buses No. 152 is exactly 5 minutes, and the interval between buses No. 251 is exactly 7 minutes. The intervals are strictly... | math | {
"style": "rule",
"ground_truth": "5/14"
} | {
"split": "train",
"index": 39207,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest natural number that is divisible by $48^{2}$ and contains only the digits 0 and 1. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11111111100000000"
} | {
"split": "train",
"index": 39208,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is known that \\( m, n, \\) and \\( k \\) are distinct natural numbers greater than 1, the number \\( \\log_{m} n \\) is rational, and additionally,\n\n$$\nk^{\\sqrt{\\log_{m} n}} = m^{\\sqrt{\\log_{n} k}}\n$$\n\nFind the minimum possible value of the sum \\( k + 5m + n \\).... | math | {
"style": "rule",
"ground_truth": "278"
} | {
"split": "train",
"index": 39209,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Add $256_{7} + 463_{7} + 132_7$. Express your answer in base $7$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1214_{7}"
} | {
"split": "train",
"index": 39210,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, each of \\( \\triangle W X Z \\) and \\( \\triangle X Y Z \\) is an isosceles right-angled triangle. The length of \\( W X \\) is \\( 6 \\sqrt{2} \\). The perimeter of quadrilateral \\( W X Y Z \\) is closest to Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 39211,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the terminal side of angle $\\alpha$ passes through point $P$((-$\tsqrt{3}$, $m$)), and $\\sin \\alpha$ = $\\frac{\\sqrt{3}}{4}$ $m$($m$($m$ne 0)), determine in which quadrant angle $\\alpha$ lies, and find the value of $\\tan \\alpha$. Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{7}}{3}"
} | {
"split": "train",
"index": 39212,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given point $M(2,0)$, draw two tangent lines $MA$ and $MB$ from $M$ to the circle $x^{2}+y^{2}=1$, where $A$ and $B$ are the tangent points. Calculate the dot product of vectors $\\overrightarrow{MA}$ and $\\overrightarrow{MB}$. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "\\dfrac{3}{2}"
} | {
"split": "train",
"index": 39213,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the equations:\n\\[\n\\frac{x}{a} + \\frac{y}{b} + \\frac{z}{c} = 4 \\quad \\text{and} \\quad \\frac{a}{x} + \\frac{b}{y} + \\frac{c}{z} = 3,\n\\]\nfind the value of \\(\\frac{x^2}{a^2} + \\frac{y^2}{b^2} + \\frac{z^2}{c^2}\\). Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 39214,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( a, b \\in \\{2, 3, \\cdots, 8\\} \\). Find the maximum value of \\(\\frac{a}{10b + a} + \\frac{b}{10a + b}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{89}{287}"
} | {
"split": "train",
"index": 39215,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence \\(\\{a_n\\}\\) is defined such that \\(a_1 = \\frac{\\pi}{6}\\) and \\(a_{n+1} = \\arctan \\left(\\sec a_n\\right)\\) for \\( n \\in \\mathbf{N}^{*}\\). Find the positive integer \\(m\\) such that\n\\[\n\\sin a_1 \\cdot \\sin a_2 \\cdots \\cdot \\sin a_m = \\frac{... | math | {
"style": "rule",
"ground_truth": "3333"
} | {
"split": "train",
"index": 39216,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(A B C D E\\) be a square pyramid of height \\(\\frac{1}{2}\\) with a square base \\(A B C D\\) of side length \\(A B = 12\\) (so \\(E\\) is the vertex of the pyramid, and the foot of the altitude from \\(E\\) to \\(A B C D\\) is the center of square \\(A B C D\\)). The f... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{144}"
} | {
"split": "train",
"index": 39217,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two strips of width 2 overlap at an angle of 60 degrees inside a rectangle of dimensions 4 units by 3 units. Find the area of the overlap, considering that the angle is measured from the horizontal line of the rectangle.\nA) $\\frac{2\\sqrt{3}}{3}$\nB) $\\frac{8\\sqrt{3}}{9}$\n... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 39218,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of the quadratic $y = ax^2 + bx + c$ is a parabola that passes through the points $(-1,7)$, $(5,7)$, and $(6,10)$. What is the $x$-coordinate of the vertex of the parabola? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 39219,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the vertex of the parabola C is O(0,0), and the focus is F(0,1).\n\n(1) Find the equation of the parabola C;\n(2) A line passing through point F intersects parabola C at points A and B. If lines AO and BO intersect line l: y = x - 2 at points M and N respectively, fi... | math | {
"style": "rule",
"ground_truth": "\\frac {8 \\sqrt {2}}{5}"
} | {
"split": "train",
"index": 39220,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In civil engineering, the stability of cylindrical columns under a load is often examined using the formula \\( L = \\frac{50T^3}{SH^2} \\), where \\( L \\) is the load in newtons, \\( T \\) is the thickness of the column in centimeters, \\( H \\) is the height of the column in... | math | {
"style": "rule",
"ground_truth": "31.25"
} | {
"split": "train",
"index": 39221,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a square \\(ABCD\\) with side length 1, determine the maximum value of \\(PA \\cdot PB \\cdot PC \\cdot PD\\) where point \\(P\\) lies inside or on the boundary of the square. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{16}"
} | {
"split": "train",
"index": 39222,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\sqrt{\\log_{3}(4x-1)} + \\sqrt{16-2^{x}}$, its domain is A. \n(1) Find the set A; \n(2) If the function $g(x) = (\\log_{2}x)^{2} - 2\\log_{2}x - 1$, and $x \\in A$, find the maximum and minimum values of the function $g(x)$ and the corresponding v... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 39223,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the first tank is $\\tfrac{3}{4}$ full of oil and the second tank is empty, while the second tank becomes $\\tfrac{5}{8}$ full after oil transfer, determine the ratio of the volume of the first tank to that of the second tank. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "\\frac{6}{5}"
} | {
"split": "train",
"index": 39224,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest common factor of 180, 240, and 300? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 39225,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Laura and her grandmother Ana just discovered that last year, their ages were divisible by 8 and that next year, their ages will be divisible by 7. Grandma Ana is not yet 100 years old. What is Laura's age? Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 39226,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A set \\( \\mathcal{S} \\) of distinct positive integers has the property that for every integer \\( x \\) in \\( \\mathcal{S}, \\) the arithmetic mean of the set of values obtained by deleting \\( x \\) from \\( \\mathcal{S} \\) is an integer. Given that 1 belongs to \\( \\mat... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 39227,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\mathbf{A}^{-1} = \\begin{pmatrix} -4 & 1 \\\\ 0 & 2 \\end{pmatrix},$ then find the inverse of $\\mathbf{A}^2.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix}16 & -2 \\\\ 0 & 4 \\end{pmatrix}"
} | {
"split": "train",
"index": 39228,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let A and B be \ffixed points in the plane with distance AB = 1. An ant walks on a straight\nline from point A to some point C in the plane and notices that the distance from itself to B\nalways decreases at any time during this walk. Compute the area of the region in the plane... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 39229,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let α and β be acute angles, and cos α = 1/7, sin(α + β) = 5√3/14. Find β. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 39230,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S_i$ be the set of all integers $n$ such that $100i\\leq n < 100(i + 1)$. For example, $S_4$ is the set ${400,401,402,\\ldots,499}$. How many of the sets $S_0, S_1, S_2, \\ldots, S_{999}$ do not contain a perfect square? Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "708"
} | {
"split": "train",
"index": 39231,
"solution": "The difference between consecutive squares is $(x + 1)^2 - x^2 = 2x + 1$, which means that all squares above $50^2 = 2500$ are more than $100$ apart.\n Then the first $26$ sets ($S_0,\\cdots S_{25}$) each have at least one perfect square because the differences bet... |
deepscaler | [
{
"role": "user",
"content": "Let \\( x, y, z \\) be nonnegative real numbers. Define:\n\\[\nA = \\sqrt{x + 3} + \\sqrt{y + 6} + \\sqrt{z + 12}, \n\\]\n\\[\nB = \\sqrt{x + 2} + \\sqrt{y + 2} + \\sqrt{z + 2}.\n\\]\nFind the minimum value of \\( A^2 - B^2 \\). Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 39232,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A literary and art team went to a nursing home for a performance. Originally, there were 6 programs planned, but at the request of the elderly, they decided to add 3 more programs. However, the order of the original six programs remained unchanged, and the added 3 programs were... | math | {
"style": "rule",
"ground_truth": "210"
} | {
"split": "train",
"index": 39233,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each value of $x,$ $f(x)$ is defined to be the minimum value of the three numbers $2x + 2,$ $\\frac{1}{2} x + 1,$ and $-\\frac{3}{4} x + 7.$ Find the maximum value of $f(x).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17}{5}"
} | {
"split": "train",
"index": 39234,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For the quadrilateral $ABCD$, it is known that $\\angle BAC = \\angle CAD = 60^{\\circ}$, and $AB + AD = AC$. Additionally, it is known that $\\angle ACD = 23^{\\circ}$. How many degrees is the angle $\\angle ABC$? Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "83"
} | {
"split": "train",
"index": 39235,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $G$: $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with an eccentricity of $\\frac{\\sqrt{6}}{3}$, a right focus at $(2 \\sqrt{2},0)$, and a line $l$ with a slope of $1$ intersecting the ellipse $G$ at points $A$ and $B$. An isosceles triangle is c... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"split": "train",
"index": 39236,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the sum:\n\\[\\sin^2 6^\\circ + \\sin^2 12^\\circ + \\sin^2 18^\\circ + \\dots + \\sin^2 174^\\circ.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 39237,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(a,\\ b,\\ c,\\ d\\) be real numbers such that \\(a + b + c + d = 10\\) and\n\\[ab + ac + ad + bc + bd + cd = 20.\\]\nFind the largest possible value of \\(d\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5 + \\sqrt{105}}{2}"
} | {
"split": "train",
"index": 39238,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It takes Clea 60 seconds to walk down an escalator when it is not operating, and only 24 seconds to walk down the escalator when it is operating. How many seconds does it take Clea to ride down the operating escalator when she just stands on it? Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 39239,
"solution": "1. **Define Variables:**\n Let $s$ be the speed of the escalator and $c$ be the speed of Clea walking down the escalator. Let $d$ represent the distance of the escalator.\n\n2. **Translate the Problem into Equations:**\n - When the escalator is not operating, C... |
deepscaler | [
{
"role": "user",
"content": "Let $Q$ be the product of the first $150$ positive odd integers. Find the largest integer $k'$ such that $Q$ is divisible by $3^{k'}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "76"
} | {
"split": "train",
"index": 39240,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $A=(6,13)$ and $B=(12,11)$ lie on circle $\\omega$ in the plane. Suppose that the tangent lines to $\\omega$ at $A$ and $B$ intersect at a point on the $x$-axis. What is the area of $\\omega$? Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "\\frac{85\\pi}{8}"
} | {
"split": "train",
"index": 39241,
"solution": "1. **Identify the midpoint of segment $AB$**:\n Given points $A=(6,13)$ and $B=(12,11)$, the midpoint $D$ of $AB$ is calculated as:\n \\[\n D = \\left(\\frac{6+12}{2}, \\frac{13+11}{2}\\right) = (9, 12).\n \\]\n\n2. **Determine the slope of line $AB$**:\n ... |
deepscaler | [
{
"role": "user",
"content": "For an arithmetic sequence {a_n} with the sum of the first n terms denoted as S_n, it is known that (a_2 - 1)^3 + 2014(a_2 - 1) = sin \\frac{2011\\pi}{3} and (a_{2013} - 1)^3 + 2014(a_{2013} - 1) = cos \\frac{2011\\pi}{6}. Determine the value of S_{2014}. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "2014"
} | {
"split": "train",
"index": 39242,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Terri produces a sequence of positive integers by following three rules. She starts with a positive integer, then applies the appropriate rule to the result, and continues in this fashion. \n\nRule 1: If the integer is less than 10, multiply it by 9.\nRule 2: If the integer i... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 39243,
"solution": "To solve the problem, we first need to understand the sequence generated by the rules provided and then determine the $98^\\text{th}$ term of the sequence starting with 98.\n\n1. **Starting the sequence**: We begin with 98 and apply the rules:\n - **98** is even ... |
deepscaler | [
{
"role": "user",
"content": "When three standard dice are tossed, the numbers $x, y, z$ are obtained. Find the probability that $xyz = 8$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{36}"
} | {
"split": "train",
"index": 39244,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\( x=2 \\), what is the value of \\( (x+2-x)(2-x-2) \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 39245,
"solution": "When \\( x=2 \\), we have \\( (x+2-x)(2-x-2)=(2+2-2)(2-2-2)=(2)(-2)=-4 \\). Alternatively, we could simplify \\( (x+2-x)(2-x-2) \\) to obtain \\( (2)(-x) \\) or \\( -2x \\) and then substitute \\( x=2 \\) to obtain a result of \\( -2(2) \\) or -4."
} |
deepscaler | [
{
"role": "user",
"content": "Admiral Ackbar needs to send a 5-character message through hyperspace to the Rebels. Each character is a lowercase letter, and the same letter may appear more than once in a message. When the message is beamed through hyperspace, the characters come out in a random order. Ackba... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 39246,
"solution": "If there is more than one distinct letter sent in the message, then there will be at most a $1/5$ chance of transmitting the right message. So the message must consist of one letter repeated five times, so there are 26 possible messages."
} |
deepscaler | [
{
"role": "user",
"content": "Find the coefficient of $x^3$ when $3(x^2 - x^3+x) +3(x +2x^3- 3x^2 + 3x^5+x^3) -5(1+x-4x^3 - x^2)$ is simplified. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 39247,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four boys bought a boat for $60. The first boy paid one half of the sum of the amounts paid by the other boys; the second boy paid one third of the sum of the amounts paid by the other boys; and the third boy paid one fourth of the sum of the amounts paid by the other boys. How... | math | {
"style": "rule",
"ground_truth": "$13"
} | {
"split": "train",
"index": 39248,
"solution": "Let the amounts paid by the first, second, third, and fourth boys be $a$, $b$, $c$, and $d$ respectively. According to the problem, we have the following equations:\n\n1. $a = \\frac{1}{2}(b+c+d)$\n2. $b = \\frac{1}{3}(a+c+d)$\n3. $c = \\frac{1}{4}(a+b+d)$\n4. $a +... |
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