data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Which of the following numbers is an odd integer, contains the digit 5, is divisible by 3, and lies between \\(12^{2}\\) and \\(13^{2}\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "165"
} | {
"split": "train",
"index": 2000,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\left\\{x_{n}\\right\\}$ be a sequence of positive real numbers. If $x_{1}=\\frac{3}{2}$ and $x_{n+1}^{2}-x_{n}^{2}=\\frac{1}{(n+2)^{2}}-\\frac{1}{n^{2}}$ for all positive integers $n$, find $x_{1}+x_{2}+\\cdots+x_{2009}$. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "\\frac{4040099}{2010}"
} | {
"split": "train",
"index": 2001,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At the rally commemorating the 60th anniversary of the Chinese people's victory in the War of Resistance against Japan, two schools each send 3 representatives to speak in turns, criticizing the heinous crimes committed by the Japanese aggressors and praising the heroic deeds o... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 2002,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Emily cycles at a constant rate of 15 miles per hour, and Leo runs at a constant rate of 10 miles per hour. If Emily overtakes Leo when he is 0.75 miles ahead of her, and she can view him in her mirror until he is 0.6 miles behind her, calculate the time in minutes it takes for... | math | {
"style": "rule",
"ground_truth": "16.2"
} | {
"split": "train",
"index": 2003,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Pyramid $OABCD$ has square base $ABCD,$ congruent edges $\\overline{OA}, \\overline{OB}, \\overline{OC},$ and $\\overline{OD},$ and $\\angle AOB=45^\\circ.$ Let $\\theta$ be the measure of the dihedral angle formed by faces $OAB$ and $OBC.$ Given that $\\cos \\theta=m+\\sqrt{n}... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 2004,
"solution": "[asy] import three; // calculate intersection of line and plane // p = point on line // d = direction of line // q = point in plane // n = normal to plane triple lineintersectplan(triple p, triple d, triple q, triple n) { return (p + dot(n,q - p)/dot(n,d)*d); } // p... |
deepscaler | [
{
"role": "user",
"content": "The real numbers \\( x_{1}, x_{2}, \\cdots, x_{2001} \\) satisfy \\( \\sum_{k=1}^{2000}\\left|x_{k}-x_{k+1}\\right| = 2001 \\). Let \\( y_{k} = \\frac{1}{k} \\left( x_{1} + x_{2} + \\cdots + x_{k} \\right) \\) for \\( k = 1, 2, \\cdots, 2001 \\). Find the maximum possible value... | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"split": "train",
"index": 2005,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that a 4-digit positive integer has only even digits (0, 2, 4, 6, 8) and is divisible by 4, calculate the number of such integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 2006,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Richard and Shreyas are arm wrestling against each other. They will play $10$ rounds, and in each round, there is exactly one winner. If the same person wins in consecutive rounds, these rounds are considered part of the same “streak”. How many possible outcomes are there in ... | math | {
"style": "rule",
"ground_truth": "932"
} | {
"split": "train",
"index": 2007,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nIn how many ways can a bamboo trunk (a non-uniform natural material) of length 4 meters be cut into three parts, the lengths of which are multiples of 1 decimeter, and from which a triangle can be formed? Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "171"
} | {
"split": "train",
"index": 2008,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a, b, c$ are positive real numbers and $\\log _{a} b+\\log _{b} c+\\log _{c} a=0$, find the value of $\\left(\\log _{a} b\\right)^{3}+\\left(\\log _{b} c\\right)^{3}+\\left(\\log _{c} a\\right)^{3}$. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 2009,
"solution": "3. Let $x=\\log _{a} b$ and $y=\\log _{b} c$; then $\\log _{c} a=-(x+y)$. Thus we want to compute the value of $x^{3}+y^{3}-(x+y)^{3}=-3 x^{2} y-3 x y^{2}=-3 x y(x+y)$. On the other hand, $-x y(x+y)=\\left(\\log _{a} b\\right)\\left(\\log _{b} c\\right)\\left(\\log ... |
deepscaler | [
{
"role": "user",
"content": "When a polynomial is divided by $-3x^5 + 10x - 11,$ what are the possible degrees of the remainder? Enter all the possible values, separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0,1,2,3,4"
} | {
"split": "train",
"index": 2010,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle has three different integer side lengths and a perimeter of 20 units. What is the maximum length of any one side? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 2011,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that \\((x_{0}, y_{0})\\) is a solution of the system:\n\\[\n\\begin{cases}\nxy = 6 \\\\\nx^2 y + xy^2 + x + y + c = 2\n\\end{cases}\n\\]\nFind the value of \\(d = x_{0}^{2} + y_{0}^{2}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "69"
} | {
"split": "train",
"index": 2012,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of the function y=sin(2x+φ) is shifted to the left by π/6 units along the x-axis, resulting in an even function graph. Determine the value of φ such that the equation 2(x + π/6) + φ = -x + 2πk is satisfied for some integer k. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{6}"
} | {
"split": "train",
"index": 2013,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$ , where we keep track of the order of the summation. For example, $f(4)=6$ because $4$ can be written as $4$ , $2+2$ , $2+1+1$ , $1+2+1$ , $1+1+2$ , and $1+1+1+1$ . Find the smallest $n$ greater than $201... | math | {
"style": "rule",
"ground_truth": "2047"
} | {
"split": "train",
"index": 2014,
"solution": "First of all, note that $f(n)$ = $\\sum_{i=0}^{k} f(n-2^{i})$ where $k$ is the largest integer such that $2^k \\le n$ . We let $f(0) = 1$ for convenience.\nFrom here, we proceed by induction, with our claim being that the only $n$ such that $f(n)$ is odd are $n$ re... |
deepscaler | [
{
"role": "user",
"content": "A square has vertices \\( P, Q, R, S \\) labelled clockwise. An equilateral triangle is constructed with vertices \\( P, T, R \\) labelled clockwise. What is the size of angle \\( \\angle RQT \\) in degrees? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "135"
} | {
"split": "train",
"index": 2015,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given triangle \\( \\triangle ABC \\) with \\( Q \\) as the midpoint of \\( BC \\), \\( P \\) on \\( AC \\) such that \\( CP = 3PA \\), and \\( R \\) on \\( AB \\) such that \\( S_{\\triangle PQR} = 2 S_{\\triangle RBQ} \\). If \\( S_{\\triangle ABC} = 300 \\), find \\( S_{\\tr... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 2016,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Complex numbers $p,$ $q,$ and $r$ are zeros of a polynomial $P(z) = z^3 + sz + t,$ and $|p|^2 + |q|^2 + |r|^2 = 325.$ The points corresponding to $p,$ $q,$ and $r$ in the complex plane form a right triangle with right angle at $q.$ Find the square of the hypotenuse, $h^2,$ of t... | math | {
"style": "rule",
"ground_truth": "487.5"
} | {
"split": "train",
"index": 2017,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A notebook containing 67 pages, numbered from 1 to 67, is renumbered such that the last page becomes the first one, the second-last becomes the second one, and so on. Determine how many pages have the same units digit in their old and new numbering. Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 2018,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the last three digits of the product of the positive roots of $\\sqrt{1995}x^{\\log_{1995}x}=x^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 2019,
"solution": "Taking the $\\log_{1995}$ (logarithm) of both sides and then moving to one side yields the quadratic equation $2(\\log_{1995}x)^2 - 4(\\log_{1995}x) + 1 = 0$. Applying the quadratic formula yields that $\\log_{1995}x = 1 \\pm \\frac{\\sqrt{2}}{2}$. Thus, the product... |
deepscaler | [
{
"role": "user",
"content": "For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\\frac{n}{180}$ terminate? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 2020,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If a polygon is formed by connecting a point on one side of the polygon to all its vertices, forming $2023$ triangles, determine the number of sides of this polygon. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2024"
} | {
"split": "train",
"index": 2021,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\theta = 25^\\circ$ be an angle such that $\\tan \\theta = \\frac{1}{6}$. Compute $\\sin^6 \\theta + \\cos^6 \\theta$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{12}"
} | {
"split": "train",
"index": 2022,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $100^a = 4$ and $100^b = 5,$ then find $20^{(1 - a - b)/(2(1 - b))}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{20}"
} | {
"split": "train",
"index": 2023,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sin (x+ \\frac{7\\pi}{4})+\\cos (x- \\frac{3\\pi}{4})$, where $x\\in R$.\n(1) Find the smallest positive period and the minimum value of $f(x)$;\n(2) Given that $f(\\alpha)= \\frac{6}{5}$, where $0 < \\alpha < \\frac{3\\pi}{4}$, find the value of $f(2... | math | {
"style": "rule",
"ground_truth": "\\frac{31\\sqrt{2}}{25}"
} | {
"split": "train",
"index": 2024,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider parallelogram $A B C D$ with $A B>B C$. Point $E$ on $\\overline{A B}$ and point $F$ on $\\overline{C D}$ are marked such that there exists a circle $\\omega_{1}$ passing through $A, D, E, F$ and a circle $\\omega_{2}$ passing through $B, C, E, F$. If $\\omega_{1}, \\o... | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 2025,
"solution": "We want to find $A D=B C=E F$. So, let $E F$ intersect $B D$ at $O$. It is clear that $\\triangle B O E \\sim \\triangle D O F$. However, we can show by angle chase that $\\triangle B X E \\sim \\triangle D Y F$ : $$\\angle B E G=\\angle A D G=\\angle C B H=\\angle ... |
deepscaler | [
{
"role": "user",
"content": "The robotics club has 30 members: 12 boys and 18 girls. A 6-person committee is chosen at random. What is the probability that the committee has at least 1 boy and at least 1 girl? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{574,287}{593,775}"
} | {
"split": "train",
"index": 2026,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The café \"Buratino\" operates 6 days a week with Mondays off. Kolya said that from April 1 to April 20, the café was open for 17 days, and from April 10 to April 30, it was open for 18 days. It is known that he made a mistake once. What was the date of the last Tuesday in Apri... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 2027,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a $\\textit{better word}$ as a sequence of letters that consists only of the letters $A$, $B$, $C$, and $D$ — some of these letters may not appear in the sequence — where $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, $C$ is never imm... | math | {
"style": "rule",
"ground_truth": "2916"
} | {
"split": "train",
"index": 2028,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 1.$ Find the maximum value of $x^3 y^2 z.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{432}"
} | {
"split": "train",
"index": 2029,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $\\angle A= \\frac {\\pi}{3}$, $BC=3$, $AB= \\sqrt {6}$, find $\\angle C=$ \\_\\_\\_\\_\\_\\_ and $AC=$ \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6} + 3\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 2030,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An archipelago consists of \\( N \\geq 7 \\) islands. Any two islands are connected by no more than one bridge. It is known that no more than 5 bridges lead from each island, and among any 7 islands, there are necessarily two that are connected by a bridge. What is the maximum ... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 2031,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 8 books on a shelf, which include a trilogy that must be selected together. In how many ways can 5 books be selected from this shelf if the order in which the books are selected does not matter? Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 2032,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\[f(x) = \\left\\{\n\\begin{array}{cl}\n\\frac{x}{21} & \\text{ if }x\\text{ is a multiple of 3 and 7}, \\\\\n3x & \\text{ if }x\\text{ is only a multiple of 7}, \\\\\n7x & \\text{ if }x\\text{ is only a multiple of 3}, \\\\\nx+3 & \\text{ if }x\\text{ is not a multiple of... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 2033,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six boys stood equally spaced on a circle of radius 40 feet. Each boy walked to all of the other non-adjacent persons on the circle, shook their hands and then returned to his original spot on the circle before the next boy started his trip to shake hands with all of the other ... | math | {
"style": "rule",
"ground_truth": "480 + 480\\sqrt{3}\\text{ feet}"
} | {
"split": "train",
"index": 2034,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\ell$ and $m$ be two non-coplanar lines in space, and let $P_{1}$ be a point on $\\ell$. Let $P_{2}$ be the point on $m$ closest to $P_{1}, P_{3}$ be the point on $\\ell$ closest to $P_{2}, P_{4}$ be the point on $m$ closest to $P_{3}$, and $P_{5}$ be the point on $\\ell$... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{39}}{4}"
} | {
"split": "train",
"index": 2035,
"solution": "Let $a$ be the answer. By taking the $z$-axis to be the cross product of these two lines, we can let the lines be on the planes $z=0$ and $z=h$, respectively. Then, by projecting onto the $xy$-plane, we get the above diagram. The projected lengths of the first four ... |
deepscaler | [
{
"role": "user",
"content": "Let \\( A = (2,0) \\) and \\( B = (8,6) \\). Let \\( P \\) be a point on the parabola \\( y^2 = 8x \\). Find the smallest possible value of \\( AP + BP \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 2036,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a factory, Erika assembles 3 calculators in the same amount of time that Nick assembles 2 calculators. Also, Nick assembles 1 calculator in the same amount of time that Sam assembles 3 calculators. How many calculators in total can be assembled by Nick, Erika, and Sam in the... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 2037,
"solution": "Erika assembling 9 calculators is the same as assembling three groups of 3 calculators. Since Erika assembles 3 calculators in the same amount of time that Nick assembles 2 calculators, then he assembles three groups of 2 calculators (that is, 6 calculators) in this... |
deepscaler | [
{
"role": "user",
"content": "A pentagon is inscribed around a circle, with the lengths of its sides being whole numbers, and the lengths of the first and third sides equal to 1. Into what segments does the point of tangency divide the second side? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 2038,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the vertices of a smaller square are midpoints of the sides of a larger square, and the larger square has an area of 144, what is the area of the smaller square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 2039,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Of the numbers 1, 2, 3, ..., 15, which number has the greatest number of divisors (the dots mean that we are including all the integers between 1 and 15)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 2040,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse C: $$\\frac {x^{2}}{4}+y^{2}=1,$$ \n(1) If a line $l$ passes through point Q $(1, \\frac {1}{2})$ and intersects the ellipse C at points A and B, find the equation of line $l$ such that Q is the midpoint of AB; \n(2) Given a fixed point M $(0, 2)$, and P is ... | math | {
"style": "rule",
"ground_truth": "\\frac {2 \\sqrt {21}}{3}"
} | {
"split": "train",
"index": 2041,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many (not necessarily positive) integer values of $n$ is the value of $4000 \\cdot \\left(\\frac{2}{5}\\right)^n$ an integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 2042,
"solution": "1. **Understanding the Expression**: We start by analyzing the expression $4000 \\cdot \\left(\\frac{2}{5}\\right)^n$. This expression can be rewritten as:\n \\[\n 4000 \\cdot \\left(\\frac{2}{5}\\right)^n = 4000 \\cdot 2^n \\cdot 5^{-n}\n \\]\n This shows t... |
deepscaler | [
{
"role": "user",
"content": "For positive integers $n$, the number of pairs of different adjacent digits in the binary (base two) representation of $n$ can be denoted as $D(n)$. Determine the number of positive integers less than or equal to 97 for which $D(n) = 2$. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 2043,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $24$ different complex numbers $z$ such that $z^{24}=1$. For how many of these is $z^6$ a real number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 2044,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 10 street lamps, and to save electricity, three of them are turned off. However, the two at the ends must not be turned off, and no two consecutive lamps should be off. Calculate the number of ways to do this. Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 2045,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At an elementary school, the students in third grade, fourth grade, and fifth grade run an average of $12$, $15$, and $10$ minutes per day, respectively. There are twice as many third graders as fourth graders, and twice as many fourth graders as fifth graders. What is the aver... | math | {
"style": "rule",
"ground_truth": "\\frac{88}{7}"
} | {
"split": "train",
"index": 2046,
"solution": "1. **Define the number of students in each grade**:\n Let the number of fifth graders be $f$. According to the problem, there are twice as many fourth graders as fifth graders, and twice as many third graders as fourth graders. Therefore, the number of fourth grad... |
deepscaler | [
{
"role": "user",
"content": "The height of a rhombus, drawn from the vertex of its obtuse angle, divides the side of the rhombus in the ratio $1:3$, measured from the vertex of its acute angle. What fraction of the area of the rhombus is occupied by the area of a circle inscribed in it? Let's think step by... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi \\sqrt{15}}{16}"
} | {
"split": "train",
"index": 2047,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers from 1 to 9 are placed in the cells of a 3x3 table so that the sum of the numbers on one diagonal is 7 and the sum on the other diagonal is 21. What is the sum of the numbers in the five shaded cells? Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 2048,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ be a point inside the equilateral triangle $ABC$ such that $6\\angle PBC = 3\\angle PAC = 2\\angle PCA$ . Find the measure of the angle $\\angle PBC$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 2049,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The set of positive even numbers $\\{2, 4, 6, \\cdots\\}$ is grouped in increasing order such that the $n$-th group has $3n-2$ numbers:\n\\[\n\\{2\\}, \\{4, 6, 8, 10\\}, \\{12, 14, 16, 18, 20, 22, 24\\}, \\cdots\n\\]\n\nDetermine which group contains the number 2018. Let's thin... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 2050,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The values of $a$ in the equation: $\\log_{10}(a^2 - 15a) = 2$ are: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20, -5"
} | {
"split": "train",
"index": 2051,
"solution": "1. **Convert the logarithmic equation to exponential form:**\n Given the equation $\\log_{10}(a^2 - 15a) = 2$, we can rewrite it in exponential form:\n \\[\n 10^2 = a^2 - 15a\n \\]\n Simplifying, we get:\n \\[\n 100 = a^2 - 15a\n \\]\n Rearranging ... |
deepscaler | [
{
"role": "user",
"content": "If the line $2x+my=2m-4$ is parallel to the line $mx+2y=m-2$, find the value of $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 2052,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiao Li and Xiao Hua are racing up the stairs. When Xiao Li reaches the 5th floor, Xiao Hua has reached the 3rd floor. At this rate, how many floors will Xiao Hua have reached when Xiao Li reaches the 25th floor? Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 2053,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A solid right prism $ABCDEF$ has a height of $16,$ as shown. Also, its bases are equilateral triangles with side length $12.$ Points $X,$ $Y,$ and $Z$ are the midpoints of edges $AC,$ $BC,$ and $DC,$ respectively. A part of the prism above is sliced off with a straight cut thro... | math | {
"style": "rule",
"ground_truth": "48+9\\sqrt{3}+3\\sqrt{91}"
} | {
"split": "train",
"index": 2054,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two tangents $AT$ and $BT$ touch a circle at $A$ and $B$ , respectively, and meet perpendicularly at $T$ . $Q$ is on $AT$ , $S$ is on $BT$ , and $R$ is on the circle, so that $QRST$ is a rectangle with $QT = 8$ and $ST = 9$ . Determine the radius of the cir... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 2055,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\dbinom{11}{9}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "55"
} | {
"split": "train",
"index": 2056,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the $8$ vertices of a cube, select $4$ vertices. The probability that these $4$ vertices lie in the same plane is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6}{35}"
} | {
"split": "train",
"index": 2057,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular piece of paper with side lengths 5 by 8 is folded along the dashed lines so that the folded flaps just touch at the corners as indicated by the dotted lines. Find the area of the resulting trapezoid. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "55/2"
} | {
"split": "train",
"index": 2058,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of permutations of \\( n \\) distinct elements \\( a_1, a_2, \\cdots, a_n \\) (where \\( n \\geqslant 2 \\)) such that \\( a_1 \\) is not in the first position and \\( a_2 \\) is not in the second position. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "32,527,596"
} | {
"split": "train",
"index": 2059,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\bigtriangleup PQR$ be a triangle with $\\angle P = 75^\\circ$ and $\\angle Q = 60^\\circ$. A regular hexagon $ABCDEF$ with side length 1 is drawn inside $\\triangle PQR$ so that side $\\overline{AB}$ lies on $\\overline{PQ}$, side $\\overline{CD}$ lies on $\\overline{QR}... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 2060,
"solution": "First, find that $\\angle R = 45^\\circ$. Draw $ABCDEF$. Now draw $\\bigtriangleup PQR$ around $ABCDEF$ such that $Q$ is adjacent to $C$ and $D$. The height of $ABCDEF$ is $\\sqrt{3}$, so the length of base $QR$ is $2+\\sqrt{3}$. Let the equation of $\\overline{RP}$... |
deepscaler | [
{
"role": "user",
"content": "Given $\\theta∈(0,\\frac{π}{2}$, $\\sin \\theta - \\cos \\theta = \\frac{\\sqrt{5}}{5}$, find the value of $\\tan 2\\theta$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{4}{3}"
} | {
"split": "train",
"index": 2061,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $b=2\\left(a\\cos B-c\\right)$. Find:<br/>\n$(1)$ The value of angle $A$;<br/>\n$(2)$ If $a\\cos C=\\sqrt{3}$ and $b=1$, find the value... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3} - 2"
} | {
"split": "train",
"index": 2062,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathcal{P}$ be a parabola that passes through the points $(0, 0)$ and $(12, 5)$ . Suppose that the directrix of $\\mathcal{P}$ takes the form $y = b$ . (Recall that a parabola is the set of points equidistant from a point called the focus and line called the dire... | math | {
"style": "rule",
"ground_truth": "7.2"
} | {
"split": "train",
"index": 2063,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = 4x - 9$ and $g(f(x)) = x^2 + 6x - 7$. Find $g(-8)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-87}{16}"
} | {
"split": "train",
"index": 2064,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The minimum distance from any integer-coordinate point on the plane to the line \\( y = \\frac{5}{3} x + \\frac{4}{5} \\) is to be determined. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{34}}{85}"
} | {
"split": "train",
"index": 2065,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n^{}_{}$ be the smallest positive integer that is a multiple of $75_{}^{}$ and has exactly $75_{}^{}$ positive integral divisors, including $1_{}^{}$ and itself. Find $\\frac{n}{75}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "432"
} | {
"split": "train",
"index": 2066,
"solution": "The prime factorization of $75 = 3^15^2 = (2+1)(4+1)(4+1)$. For $n$ to have exactly $75$ integral divisors, we need to have $n = p_1^{e_1-1}p_2^{e_2-1}\\cdots$ such that $e_1e_2 \\cdots = 75$. Since $75|n$, two of the prime factors must be $3$ and $5$. To minimize $... |
deepscaler | [
{
"role": "user",
"content": "\nUsing the property that in finding the limit of the ratio of two infinitesimals, they can be replaced with their equivalent infinitesimals (property II), find the following limits:\n1) \\(\\lim _{x \\rightarrow 0} \\frac{\\sin 4 x}{\\sin 3 x}\\)\n2) \\(\\lim _{x \\rightarrow ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{10}"
} | {
"split": "train",
"index": 2067,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The store bought a batch of New Year cards at 0.21 yuan each and sold them for a total of 14.57 yuan. If each card was sold at the same price and did not exceed twice the purchase price, how much profit did the store make? Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "4.7"
} | {
"split": "train",
"index": 2068,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate or simplify: \n \\\\((1)\\\\dfrac{\\\\sqrt{1-2\\\\sin {15}^{\\\\circ}\\\\cos {15}^{\\\\circ}}}{\\\\cos {15}^{\\\\circ}-\\\\sqrt{1-\\\\cos^2 {165}^{\\\\circ}}}\\\\)； \n \\\\((2)\\\\)Given \\\\(| \\\\vec{a} |=4\\\\), \\\\(| \\\\vec{b} |=2\\\\), and the angle between \\... | math | {
"style": "rule",
"ground_truth": "2\\\\sqrt{3}"
} | {
"split": "train",
"index": 2069,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph shows the total distance Sam drove from 6 a.m to 11 a.m. How many miles per hour is the car's average speed for the period from 6 a.m. to 11 a.m.?\n\n[asy]\nunitsize(0.2inch);\ndraw((0,0)--(5.5,0));\ndraw((0,0)--(0,8.5));\ndraw((1,0)--(1,8.5));\ndraw((2,0)--(2... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 2070,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $y=ax^{2}+bx+c$, where $a$, $b$, $c\\in R$.<br/>$(Ⅰ)$ If $a \\gt b \\gt c$ and $a+b+c=0$, and the distance between the graph of this function and the $x$-axis is $l$, find the range of values for $l$;<br/>$(Ⅱ)$ If $a \\lt b$ and the solution set of the inequa... | math | {
"style": "rule",
"ground_truth": "6+4\\sqrt{3}"
} | {
"split": "train",
"index": 2071,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x$ and $y$ are positive integers such that $xy - 5x + 6y = 119$, what is the minimal possible value of $|x - y|$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "77"
} | {
"split": "train",
"index": 2072,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It costs 5 cents to copy 3 pages. How many pages can you copy for $\\$20$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1200"
} | {
"split": "train",
"index": 2073,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Roger collects the first 18 U.S. state quarters released in the order that the states joined the union. Five states joined the union during the decade 1790 through 1799. What fraction of Roger's 18 quarters represents states that joined the union during this decade? Express you... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{18}"
} | {
"split": "train",
"index": 2074,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The center of a balloon is observed by two ground observers at angles of elevation of $45^{\\circ}$ and $22.5^{\\circ}$, respectively. The first observer is to the south, and the second one is to the northwest of the point directly under the balloon. The distance between the tw... | math | {
"style": "rule",
"ground_truth": "500"
} | {
"split": "train",
"index": 2075,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the set $H$ defined by the points $(x,y)$ with integer coordinates, $2\\le|x|\\le8$, $2\\le|y|\\le8$, calculate the number of squares of side at least $5$ that have their four vertices in $H$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 2076,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The odd function $y=f(x)$ has a domain of $\\mathbb{R}$, and when $x \\geq 0$, $f(x) = 2x - x^2$. If the range of the function $y=f(x)$, where $x \\in [a, b]$, is $[\\frac{1}{b}, \\frac{1}{a}]$, then the minimum value of $b$ is ______. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 2077,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two parabolas are the graphs of the equations $y=2x^2-10x-10$ and $y=x^2-4x+6$. Find all points where they intersect. List the points in order of increasing $x$-coordinate, separated by semicolons. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(8,38)"
} | {
"split": "train",
"index": 2078,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of two numbers $x$ and $y$ is 399, and the value of the fraction $\\frac{x}{y}$ is 0.9. What is the value of $y - x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 2079,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be real numbers randomly (and independently) chosen from the range $[0,1]$. Find the probability that $a, b$ and 1 form the side lengths of an obtuse triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi-2}{4}"
} | {
"split": "train",
"index": 2080,
"solution": "We require $a+b>1$ and $a^{2}+b^{2}<1$. Geometrically, this is the area enclosed in the quarter-circle centered at the origin with radius 1, not including the area enclosed by $a+b<1$ (an isosceles right triangle with side length 1). As a result, our desired probabi... |
deepscaler | [
{
"role": "user",
"content": "How many units long is a segment whose endpoints are (2,3) and (7,15)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 2081,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all the solutions to\n\\[\\frac{1}{x^2 + 11x - 8} + \\frac{1}{x^2 + 2x - 8} + \\frac{1}{x^2 - 13x - 8} = 0.\\]Enter all the solutions, separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8,1,-1,-8"
} | {
"split": "train",
"index": 2082,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In parallelogram ABCD, $\\angle BAD=60^\\circ$, $AB=1$, $AD=\\sqrt{2}$, and P is a point inside the parallelogram such that $AP=\\frac{\\sqrt{2}}{2}$. If $\\overrightarrow{AP}=\\lambda\\overrightarrow{AB}+\\mu\\overrightarrow{AD}$ ($\\lambda,\\mu\\in\\mathbb{R}$), then the maxi... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{3}"
} | {
"split": "train",
"index": 2083,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of two consecutive page numbers is $2,156.$ What are the sum of these two page numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "93"
} | {
"split": "train",
"index": 2084,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The perimeter of a square with side length $y$ inches is equal to the circumference of a circle with radius 5 centimeters. If 1 inch equals 2.54 centimeters, what is the value of $y$ in inches? Express your answer as a decimal to the nearest hundredth. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "3.09"
} | {
"split": "train",
"index": 2085,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all $t$ such that $x-t$ is a factor of $4x^2 + 11x - 3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 2086,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The minimum value of $\\frac{b^{2}+1}{\\sqrt{3}a}$ is what occurs when the eccentricity of the hyperbola $\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1$ is 2. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {4 \\sqrt {3}}{3}"
} | {
"split": "train",
"index": 2087,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a closed right triangular prism ABC-A<sub>1</sub>B<sub>1</sub>C<sub>1</sub> there is a sphere with volume $V$. If $AB \\perp BC$, $AB=6$, $BC=8$, and $AA_{1}=3$, then the maximum value of $V$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "\\frac{9\\pi}{2}"
} | {
"split": "train",
"index": 2088,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, $A$ and $B(20,0)$ lie on the $x$-axis and $C(0,30)$ lies on the $y$-axis such that $\\angle A C B=90^{\\circ}$. A rectangle $D E F G$ is inscribed in triangle $A B C$. Given that the area of triangle $C G F$ is 351, calculate the area of the rectangle $D E F G$.... | math | {
"style": "rule",
"ground_truth": "468"
} | {
"split": "train",
"index": 2089,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area of the region enclosed by $x^2 + y^2 = |x| - |y|$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"split": "train",
"index": 2090,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four tour guides are leading eight tourists. Each tourist must choose one of the guides, however, each guide must take at least two tourists. How many different groupings of guides and tourists are possible? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "105"
} | {
"split": "train",
"index": 2091,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ben throws four identical darts. Each hits one of four identical dartboards on the wall. After throwing the four darts, he lists the number of darts that hit each board, from greatest to least. How many different lists are possible? Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 2092,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}=(\\sin x,\\frac{3}{2})$ and $\\overrightarrow{b}=(\\cos x,-1)$. \n$(1)$ When $\\overrightarrow{a} \\parallel \\overrightarrow{b}$, find the value of $\\sin 2x$. \n$(2)$ Find the minimum value of $f(x)=(\\overrightarrow{a}+\\overrightarrow{b}... | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 2093,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Many states use a sequence of three letters followed by a sequence of three digits as their standard license-plate pattern. Given that each three-letter three-digit arrangement is equally likely, the probability that such a license plate will contain at least one palindrome (a ... | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 2094,
"solution": "Note that we can pick the first and second letters/numbers freely with one choice left for the last letter/number for there to be a palindrome. Thus, the probability of no palindrome is \\[\\frac{25}{26}\\cdot \\frac{9}{10}=\\frac{45}{52}\\] thus we have $1-\\frac{4... |
deepscaler | [
{
"role": "user",
"content": "\nIn the isosceles trapezoid \\( KLMN \\), the base \\( KN \\) is equal to 9, and the base \\( LM \\) is equal to 5. Points \\( P \\) and \\( Q \\) lie on the diagonal \\( LN \\), with point \\( P \\) located between points \\( L \\) and \\( Q \\), and segments \\( KP \\) and \... | math | {
"style": "rule",
"ground_truth": "7\\sqrt{21}"
} | {
"split": "train",
"index": 2095,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number $101$ is the smallest three-digit palindromic prime. What is the second-smallest one? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "131"
} | {
"split": "train",
"index": 2096,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\[f(n)=\\dfrac{5+3\\sqrt{5}}{10}\\left(\\dfrac{1+\\sqrt{5}}{2}\\right)^n+\\dfrac{5-3\\sqrt{5}}{10}\\left(\\dfrac{1-\\sqrt{5}}{2}\\right)^n.\\]\nThen $f(n+1)-f(n-1)$, expressed in terms of $f(n)$, equals: Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "f(n)"
} | {
"split": "train",
"index": 2097,
"solution": "1. **Expression for $f(n+1)$ and $f(n-1)$**:\n - We start by calculating $f(n+1)$:\n \\[\n f(n+1) = \\dfrac{5+3\\sqrt{5}}{10}\\left(\\dfrac{1+\\sqrt{5}}{2}\\right)^{n+1}+\\dfrac{5-3\\sqrt{5}}{10}\\left(\\dfrac{1-\\sqrt{5}}{2}\\right)^{n+1}\n \\]\n ... |
deepscaler | [
{
"role": "user",
"content": "In Chinese ancient music, the pentatonic scale is the basic scale, hence the idiom \"五音不全\" (meaning \"lacking in musical talent\"). The five notes of the pentatonic scale in Chinese ancient music are: 宫 (gong), 商 (shang), 角 (jue), 徵 (zhi), 羽 (yu). If these five notes are all u... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 2098,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrappi... | math | {
"style": "rule",
"ground_truth": "2(w+h)^2"
} | {
"split": "train",
"index": 2099,
"solution": "1. **Understanding the Problem Setup**: We have a square sheet of wrapping paper and a box with a square base of side $w$ and height $h$. The box is placed such that its base vertices lie on the midlines of the wrapping paper. The wrapping paper is folded up to meet... |
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