data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Which fractions $ \\dfrac{p}{q},$ where $p,q$ are positive integers $< 100$, is closest to $\\sqrt{2} ?$ Find all digits after the point in decimal representation of that fraction which coincide with digits in decimal representation of $\\sqrt{2}$ (without using any table). Let... | math | {
"style": "rule",
"ground_truth": "\\frac{99}{70}"
} | {
"split": "train",
"index": 4700,
"solution": "\nWe are tasked with finding the fraction \\(\\frac{p}{q}\\), where \\( p, q \\) are positive integers less than 100, that is closest to \\(\\sqrt{2}\\). Additionally, we aim to determine how many digits after the decimal point coincide between this fraction and \\(... |
deepscaler | [
{
"role": "user",
"content": "Given a rectangular yard containing two congruent isosceles right triangles in the form of flower beds and a trapezoidal remainder, with the parallel sides of the trapezoid having lengths $15$ and $25$ meters. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"split": "train",
"index": 4701,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all functions $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ such that $$f(f(x)+y)+xf(y)=f(xy+y)+f(x)$$ for reals $x, y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = x \\text{ or } f(x) = 0"
} | {
"split": "train",
"index": 4702,
"solution": "\nLet \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) be a function satisfying the functional equation:\n\n\\[\nf(f(x) + y) + x f(y) = f(xy + y) + f(x)\n\\]\n\nfor all real numbers \\( x \\) and \\( y \\).\n\n### Step 1: Initial Substitution\n\nStart by substituting... |
deepscaler | [
{
"role": "user",
"content": "How many points does a sports team earn for 9 wins, 3 losses, and 4 ties, if they earn 2 points for each win, 0 points for each loss, and 1 point for each tie? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 4703,
"solution": "The team earns 2 points for each win, so 9 wins earn $2 \\times 9=18$ points. The team earns 0 points for each loss, so 3 losses earn 0 points. The team earns 1 point for each tie, so 4 ties earn 4 points. In total, the team earns $18+0+4=22$ points."
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer that is both an integer power of 7 and is not a palindrome. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2401"
} | {
"split": "train",
"index": 4704,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to put 4 balls in 3 boxes if the balls are distinguishable and the boxes are distinguishable? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 4705,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three workshops A, B, and C of a factory produced the same kind of product, with quantities of 120, 60, and 30, respectively. To determine if there are significant differences in product quality, a stratified sampling method was used to take a sample of size n for inspection, w... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 4706,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $U = \\{2, 4, 3-a^2\\}$ and $P = \\{2, a^2+2-a\\}$. Given that the complement of $P$ in $U$, denoted as $\\complement_U P$, is $\\{-1\\}$, find the value of $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "a = 2"
} | {
"split": "train",
"index": 4707,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate plane $xOy$, a circle with center $C(1,1)$ is tangent to the $x$-axis and $y$-axis at points $A$ and $B$, respectively. Points $M$ and $N$ lie on the line segments $OA$ and $OB$, respectively. If $MN$ is tangent to circle $C$, find the minimum value ... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2} - 2"
} | {
"split": "train",
"index": 4708,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfy $|\\overrightarrow{a}|=2$, $|\\overrightarrow{b}|=1$ and $(\\overrightarrow{a} + \\overrightarrow{b}) \\perp \\overrightarrow{b}$, calculate the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$. ... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"split": "train",
"index": 4709,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest three-digit integer that is divisible by each of its digits and the sum of the digits is divisible by 6. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "936"
} | {
"split": "train",
"index": 4710,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The letter T is formed by placing a $2\\:\\text{inch}\\!\\times\\!6\\:\\text{inch}$ rectangle vertically and a $3\\:\\text{inch}\\!\\times\\!2\\:\\text{inch}$ rectangle horizontally on top of the vertical rectangle at its middle, as shown. What is the perimeter of this T, in in... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 4711,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{P}$ be the matrix for projecting onto the vector $\\begin{pmatrix} 4 \\\\ -7 \\end{pmatrix}.$ Find $\\det \\mathbf{P}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 4712,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When triangle $EFG$ is rotated by an angle $\\arccos(_{1/3})$ around point $O$, which lies on side $EG$, vertex $F$ moves to vertex $E$, and vertex $G$ moves to point $H$, which lies on side $FG$. Find the ratio in which point $O$ divides side $EG$. Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "3:1"
} | {
"split": "train",
"index": 4713,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A farmer bought $749$ sheep. He sold $700$ of them for the price paid for the $749$ sheep. The remaining $49$ sheep were sold at the same price per head as the other $700$. Based on the cost, the percent gain on the entire transaction is: Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 4714,
"solution": "1. **Define the cost and revenue variables:**\n Let the cost of each sheep be $x$ dollars. Therefore, the total cost for $749$ sheep is $749x$ dollars.\n\n2. **Calculate the revenue from selling 700 sheep:**\n The farmer sold $700$ sheep for the same amount he p... |
deepscaler | [
{
"role": "user",
"content": "A marathon is $26$ miles and $400$ yards. One mile equals $1760$ yards.\n\nMark has run fifteen marathons in his life. If the total distance Mark covered in these marathons is $m$ miles and $y$ yards, where $0 \\leq y < 1760$, what is the value of $y$? Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "720"
} | {
"split": "train",
"index": 4715,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\overrightarrow{a}$ and $\\overrightarrow{b}$ are non-zero vectors and $(\\overrightarrow{a} - 6\\overrightarrow{b}) \\perp \\overrightarrow{a}$, $(2\\overrightarrow{a} - 3\\overrightarrow{b}) \\perp \\overrightarrow{b}$, find the angle between $\\overrightarrow{a}... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 4716,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Complex numbers \\(a\\), \\(b\\), \\(c\\) form an equilateral triangle with side length 24 in the complex plane. If \\(|a + b + c| = 48\\), find \\(|ab + ac + bc|\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "768"
} | {
"split": "train",
"index": 4717,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that\n$$\n\\begin{array}{l}\na + b + c = 5, \\\\\na^2 + b^2 + c^2 = 15, \\\\\na^3 + b^3 + c^3 = 47.\n\\end{array}\n$$\nFind the value of \\((a^2 + ab + b^2)(b^2 + bc + c^2)(c^2 + ca + a^2)\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "625"
} | {
"split": "train",
"index": 4718,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the maximal size of a set of positive integers with the following properties: $1.$ The integers consist of digits from the set $\\{ 1,2,3,4,5,6\\}$ . $2.$ No digit occurs more than once in the same integer. $3.$ The digits in each integer are in increasing order. ... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 4719,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathcal{O}$ be a regular octahedron. How many lines are there such that a rotation of at most $180^{\\circ}$ around these lines maps $\\mathcal{O}$ onto itself? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 4720,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest value of $x$ that satisfies the equation $|3x+7|=26$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-11"
} | {
"split": "train",
"index": 4721,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The second hand on a clock is 10 cm long. How far in centimeters does the tip of this second hand travel during a period of 15 minutes? Express your answer in terms of $\\pi$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "300\\pi"
} | {
"split": "train",
"index": 4722,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest natural number \\( n \\) for which the product of the numbers \\( n, n+1, n+2, \\ldots, n+20 \\) is divisible by the square of one of these numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 4723,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest value that the expression takes $x^4 + y^4 - x^2y - xy^2$, for positive numbers $x$ and $y$ satisfying $x + y \\le 1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{8}"
} | {
"split": "train",
"index": 4724,
"solution": "\nWe wish to find the minimum value of the expression \\( x^4 + y^4 - x^2y - xy^2 \\) subject to the constraint \\( x + y \\leq 1 \\) where \\( x \\) and \\( y \\) are positive real numbers.\n\nFirst, consider using the Lagrange multipliers method to incorporate the... |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a} = (x, -3)$, $\\overrightarrow{b} = (-2, 1)$, $\\overrightarrow{c} = (1, y)$ on a plane. If $\\overrightarrow{a}$ is perpendicular to $\\overrightarrow{b} - \\overrightarrow{c}$, and $\\overrightarrow{b}$ is parallel to $\\overrightarrow{a} + \... | math | {
"style": "rule",
"ground_truth": "-\\sqrt{5}"
} | {
"split": "train",
"index": 4725,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The complex numbers $z$ and $w$ satisfy $z^{13} = w,$ $w^{11} = z,$ and the imaginary part of $z$ is $\\sin{\\frac{m\\pi}{n}}$, for relatively prime positive integers $m$ and $n$ with $m<n.$ Find $n.$ Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "71"
} | {
"split": "train",
"index": 4726,
"solution": "Substituting the first equation into the second, we find that $(z^{13})^{11} = z$ and thus $z^{143} = z.$ We know that $z \\neq 0,$ because we are given the imaginary part of $z,$ so we can divide by $z$ to get $z^{142} = 1.$ So, $z$ must be a $142$nd root of unity,... |
deepscaler | [
{
"role": "user",
"content": "How many distinct four-digit positive integers are there such that the product of their digits equals 18? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 4727,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jose invested $\\$50,\\!000$ for $2$ years at an annual interest rate of $4$ percent compounded yearly. Patricia invested $\\$50,\\!000$ for the same period of time, at the same interest rate, but the interest was compounded quarterly. To the nearest dollar, how much more money... | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 4728,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ such that a cube with sides of length $n$ can be divided up into exactly $2007$ smaller cubes, each of whose sides is of integer length. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 4729,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$, $b$, and $c$ be the sides of a triangle with angles $\\alpha$, $\\beta$, and $\\gamma$ opposite them respectively. Suppose $a^2 + b^2 = 9c^2$. Find the value of\n\\[\\frac{\\tan \\gamma}{\\tan \\alpha + \\tan \\beta}.\\] Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 4730,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $F_{1}$ and $F_{2}$ are the two foci of the ellipse $\\frac{x^{2}}{20} + \\frac{y^{2}}{4} = 1$, a line passing through $F_{2}$ intersects the ellipse at points $A$ and $B$. If $|F_{1}A| + |F_{1}B| = 5\\sqrt{5}$, then $|AB| = $ ______. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{5}"
} | {
"split": "train",
"index": 4731,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Expand and simplify the expression $-(4-d)(d+3(4-d))$. What is the sum of the coefficients of the expanded form? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-30"
} | {
"split": "train",
"index": 4732,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all ordered pairs $(a, b)$ of complex numbers with $a^{2}+b^{2} \\neq 0, a+\\frac{10b}{a^{2}+b^{2}}=5$, and $b+\\frac{10a}{a^{2}+b^{2}}=4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(1,2),(4,2),\\left(\\frac{5}{2}, 2 \\pm \\frac{3}{2} i\\right)"
} | {
"split": "train",
"index": 4733,
"solution": "First, it is easy to see that $ab \\neq 0$. Thus, we can write $\\frac{5-a}{b}=\\frac{4-b}{a}=\\frac{10}{a^{2}+b^{2}}$. Then, we have $\\frac{10}{a^{2}+b^{2}}=\\frac{4a-ab}{a^{2}}=\\frac{5b-ab}{b^{2}}=\\frac{4a+5b-2ab}{a^{2}+b^{2}}$. Therefore, $4a+5b-2ab=10$, so $(... |
deepscaler | [
{
"role": "user",
"content": "Find the smallest composite number that has no prime factors less than 15. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "289"
} | {
"split": "train",
"index": 4734,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three fair coins are to be tossed once. For each head that results, one fair die is to be rolled. Calculate the probability that the sum of the die rolls is odd. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{16}"
} | {
"split": "train",
"index": 4735,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to arrange the letters of the word $\\text{B}_1\\text{A}_1\\text{N}_1\\text{A}_2\\text{N}_2\\text{A}_3\\text{B}_2$, where three A's, two N's, and two B's are all considered different within each letter group but identical between groups? Let's think step... | math | {
"style": "rule",
"ground_truth": "210"
} | {
"split": "train",
"index": 4736,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the numbers \\( x, y, z \\in [0, \\pi] \\), find the minimum value of the expression\n\n$$\nA = \\cos (x - y) + \\cos (y - z) + \\cos (z - x)\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 4737,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the function $f\\left(x\\right)$ is an even function on $R$, and $f\\left(x+2\\right)$ is an odd function. If $f\\left(0\\right)=1$, then $f\\left(1\\right)+f\\left(2\\right)+\\ldots +f\\left(2023\\right)=\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 4738,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the circle $\\odot C: x^{2}+y^{2}-2 x-2 y=-1$, points $A(2 a, 0)$ and B(0,2 b), where $a > 1$ and $b > 1$, determine the minimum area of triangle $\\triangle A O B$ (where $O$ is the origin) when the circle $\\odot C$ is tangent to the line $AB$. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "3 + 2\\sqrt{2}"
} | {
"split": "train",
"index": 4739,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The integer solution for the inequality $|2x-m|\\leq1$ with respect to $x$ is uniquely $3$ ($m$ is an integer).\n(I) Find the value of the integer $m$;\n(II) Given that $a, b, c \\in R$, if $4a^4+4b^4+4c^4=m$, find the maximum value of $a^2+b^2+c^2$. Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 4740,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ is a right triangle with $AC = 7,$ $BC = 24,$ and right angle at $C.$ Point $M$ is the midpoint of $AB,$ and $D$ is on the same side of line $AB$ as $C$ so that $AD = BD = 15.$ Given that the area of triangle $CDM$ may be expressed as $\\frac {m\\sqrt {n}}{p},$ w... | math | {
"style": "rule",
"ground_truth": "578"
} | {
"split": "train",
"index": 4741,
"solution": "We use the Pythagorean Theorem on $ABC$ to determine that $AB=25.$\n Let $N$ be the orthogonal projection from $C$ to $AB.$ Thus, $[CDM]=\\frac{(DM)(MN)} {2}$, $MN=AM-AN$, and $[ABC]=\\frac{24 \\cdot 7} {2} =\\frac{25 \\cdot (CN)} {2}.$\n From the third equation, we... |
deepscaler | [
{
"role": "user",
"content": "Five people are at a party. Each pair of them are friends, enemies, or frenemies (which is equivalent to being both friends and enemies). It is known that given any three people $A, B, C$ : - If $A$ and $B$ are friends and $B$ and $C$ are friends, then $A$ and $C$ are friends; ... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 4742,
"solution": "If $A$ and $B$ are frenemies, then regardless of whether another person $C$ is friends or enemies with $A$, $C$ will have to be frenemies with $B$ and vice versa. Therefore, if there is one pair of frenemies then all of them are frenemies with each other, and there ... |
deepscaler | [
{
"role": "user",
"content": "Isabella's house has $3$ bedrooms. Each bedroom is $12$ feet long, $10$ feet wide, and $8$ feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy $60$ square feet in each bedroom. How many square feet of walls must ... | math | {
"style": "rule",
"ground_truth": "876"
} | {
"split": "train",
"index": 4743,
"solution": "1. **Calculate the area of the walls in one bedroom**: \n Each bedroom has four walls, two pairs of opposite walls. Each pair consists of:\n - Two walls of dimensions $12$ feet (length) by $8$ feet (height).\n - Two walls of dimensions $10$ feet (width) by $8$... |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive four-digit number that is divisible by each of the four smallest prime numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1050"
} | {
"split": "train",
"index": 4744,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$ with common difference $d \\neq 0$, and its first term $a_1 = d$. The sum of the first $n$ terms of the sequence $\\{a_n^2\\}$ is denoted as $S_n$. Additionally, there is a geometric sequence $\\{b_n\\}$ with a common ratio $q$ that is a... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 4745,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\( b\\left[\\frac{1}{1 \\times 3}+\\frac{1}{3 \\times 5}+\\cdots+\\frac{1}{1999 \\times 2001}\\right]=2 \\times\\left[\\frac{1^{2}}{1 \\times 3}+\\frac{2^{2}}{3 \\times 5}+\\cdots+\\frac{1000^{2}}{1999 \\times 2001}\\right] \\), find the value of \\( b \\). Let's th... | math | {
"style": "rule",
"ground_truth": "1001"
} | {
"split": "train",
"index": 4746,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ed has five identical green marbles, and a large supply of identical red marbles. He arranges the green marbles and some of the red ones in a row and finds that the number of marbles whose right hand neighbor is the same color as themselves is equal to the number of marbles who... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 4747,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an isosceles trapezoid, the parallel bases have lengths $\\log 3$ and $\\log 192$, and the altitude to these bases has length $\\log 16$. The perimeter of the trapezoid can be written in the form $\\log 2^p 3^q$, where $p$ and $q$ are positive integers. Find $p + q$. Let's t... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 4748,
"solution": "Call the trapezoid $ABCD$ with $AB$ as the smaller base and $CD$ as the longer. Let the point where an altitude intersects the larger base be $E$, where $E$ is closer to $D$.\n Subtract the two bases and divide to find that $ED$ is $\\log 8$. The altitude can be exp... |
deepscaler | [
{
"role": "user",
"content": "Let $f$ be a random permutation on $\\{1, 2, \\dots, 100\\}$ satisfying $f(1) > f(4)$ and $f(9)>f(16)$ . The probability that $f(1)>f(16)>f(25)$ can be written as $\\frac mn$ where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$ .\n\nNote: I... | math | {
"style": "rule",
"ground_truth": "124"
} | {
"split": "train",
"index": 4749,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine all positive integers $n$ for which there exists an integer $m$ such that ${2^{n}-1}$ is a divisor of ${m^{2}+9}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "n = 2^k"
} | {
"split": "train",
"index": 4750,
"solution": "\nWe want to determine all positive integers \\( n \\) for which there exists an integer \\( m \\) such that \\( 2^n - 1 \\mid m^2 + 9 \\).\n\nTo solve this problem, we start by expressing the divisibility condition explicitly:\n\n\\[\n2^n - 1 \\mid m^2 + 9 \\quad \... |
deepscaler | [
{
"role": "user",
"content": "From the set {1, 2, 3, ..., 14}, select \\(a_{1}, a_{2}, a_{3}\\) in increasing order such that \\(a_{2} - a_{1} \\geq 3\\) and \\(a_{3} - a_{2} \\geq 3\\). Find the total number of different ways to select these numbers. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 4751,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the plane rectangular coordinate system $(xOy)$, with the origin as the pole and the positive semi-axis of $x$ as the polar axis, establish a polar coordinate system with the same unit of length. The parametric equation of line $l$ is $\\begin{cases}x=2+\\frac{\\sqrt{2}}{2}t... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 4752,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using a $4 \\times 4$ grid where points are spaced equally at 1 unit apart both horizontally and vertically, how many rectangles are there whose four vertices are points on this grid? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "101"
} | {
"split": "train",
"index": 4753,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In acute triangle \\( ABC \\), \\( M \\) and \\( N \\) are the midpoints of sides \\( AB \\) and \\( BC \\), respectively. The tangents to the circumcircle of triangle \\( BMN \\) at \\( M \\) and \\( N \\) meet at \\( P \\). Suppose that \\( AP \\) is parallel to \\( BC \\), \... | math | {
"style": "rule",
"ground_truth": "20\\sqrt{2}"
} | {
"split": "train",
"index": 4754,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $\\cot 45^\\circ.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 4755,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p$, $q$, $r$, $s$, and $t$ be positive integers with $p+q+r+s+t=2025$ and let $N$ be the largest of the sums $p+q$, $q+r$, $r+s$, and $s+t$. Determine the smallest possible value of $N$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "676"
} | {
"split": "train",
"index": 4756,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the product of the first $n$ terms of the sequence $\\{a_{n}\\}$ is $T_{n}$, where ${a_n}=\\frac{n}{{2n-5}}$, determine the maximum value of $T_{n}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{3}"
} | {
"split": "train",
"index": 4757,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C$ is $\\rho^{2}-2\\rho\\cos \\theta-4\\rho\\sin \\theta+4=0$, and the equation of line $l$ is $x-y-1=0$. \n$(1)$ Write the p... | math | {
"style": "rule",
"ground_truth": "1+ \\sqrt{2}"
} | {
"split": "train",
"index": 4758,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $P$, $Q$, $R$, and $S$ lie in the plane of the square $EFGH$ so that $EPF$, $FQG$, $GRH$, and $HSE$ are equilateral triangles inside square $EFGH$. If the side length of square $EFGH$ is 8, find the area of quadrilateral $PQRS$. Assume that the vertices $P$, $Q$, $R$, an... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 4759,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many non-empty subsets $T$ of $\\{1,2,3,\\ldots,17\\}$ have the following two properties?\n$(1)$ No two consecutive integers belong to $T$.\n$(2)$ If $T$ contains $k$ elements, then $T$ contains no number less than $k+1$. Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "594"
} | {
"split": "train",
"index": 4760,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\frac{x}{9}, \\frac{y}{15}, \\frac{z}{14}$ are all in their simplest forms and their product is $\\frac{1}{6}$, find the value of $x+y+z$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 4761,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( \\triangle ABC \\), \\( BC=a \\), \\( AC=b \\), \\( AB=c \\), and \\( \\angle C = 90^{\\circ} \\). \\( CD \\) and \\( BE \\) are two medians of \\( \\triangle ABC \\), and \\( CD \\perp BE \\). Express the ratio \\( a:b:c \\) in simplest form. Let's think step b... | math | {
"style": "rule",
"ground_truth": "1 : \\sqrt{2} : \\sqrt{3}"
} | {
"split": "train",
"index": 4762,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all positive integers $n$ such that there are $k \\geq 2$ positive rational numbers $a_1, a_2, \\ldots, a_k$ satisfying $a_1 + a_2 + \\ldots + a_k = a_1 \\cdot a_2 \\cdots a_k = n.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4 \\text{ or } n \\geq 6"
} | {
"split": "train",
"index": 4763,
"solution": "\nWe are tasked with finding all positive integers \\( n \\) for which there exist \\( k \\geq 2 \\) positive rational numbers \\( a_1, a_2, \\ldots, a_k \\) satisfying the conditions:\n\n\\[\na_1 + a_2 + \\cdots + a_k = a_1 \\cdot a_2 \\cdots a_k = n.\n\\]\n\nTo fi... |
deepscaler | [
{
"role": "user",
"content": "If 2 cards from a standard deck are selected randomly, what is the probability that either two kings or at least 1 ace occurs? (There are 4 Aces, 4 kings and 52 total cards in a standard deck.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{13}"
} | {
"split": "train",
"index": 4764,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$, respectively. If $(\\sqrt{3}b-c)\\cos A=a\\cos C$, then $\\cos A=$_______. 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": 4765,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$\\_$\\_$\\_$:20=24÷$\\_$\\_$\\_$=80%=$\\_$\\_$\\_$(fill in the blank with a fraction)=$\\_$\\_$\\_$(fill in the blank with a decimal) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.8"
} | {
"split": "train",
"index": 4766,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ordered pairs \\((b, g)\\) of positive integers with \\(4 \\leq b \\leq g \\leq 2007\\) are there such that when \\(b\\) black balls and \\(g\\) gold balls are randomly arranged in a row, the probability that the balls on each end have the same colour is \\(\\frac{1}{2... | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 4767,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nBusinessmen Ivanov, Petrov, and Sidorov decided to create a car company. Ivanov bought 70 identical cars for the company, Petrov bought 40 identical cars, and Sidorov contributed 44 million rubles to the company. It is known that Ivanov and Petrov can share the money among th... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 4768,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Inside a right triangle \\(ABC\\) with hypotenuse \\(AC\\), a point \\(M\\) is chosen such that the areas of triangles \\(ABM\\) and \\(BCM\\) are one-third and one-quarter of the area of triangle \\(ABC\\) respectively. Find \\(BM\\) if \\(AM = 60\\) and \\(CM = 70\\). If the ... | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 4769,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jane is considering buying a sweater priced at $50. A store is offering a 10% discount on the sweater. After applying the discount, Jane needs to pay a state sales tax of 7.5%, and a local sales tax of 7%. Calculate the difference between the state and local sales taxes that Ja... | math | {
"style": "rule",
"ground_truth": "0.225"
} | {
"split": "train",
"index": 4770,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagram shows a square and a regular decagon that share an edge. One side of the square is extended to meet an extended edge of the decagon. What is the value of \\( x \\)?\n\nA) 15 \nB) 18 \nC) 21 \nD) 24 \nE) 27 Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 4771,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest possible value of \\( n \\) such that there exist \\( n \\) consecutive positive integers whose sum is equal to 2010. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 4772,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The inclination angle of the line $\\sqrt{3}x+y-1=0$ is ____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"split": "train",
"index": 4773,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": " $25$ checkers are placed on $25$ leftmost squares of $1 \\times N$ board. Checker can either move to the empty adjacent square to its right or jump over adjacent right checker to the next square if it is empty. Moves to the left are not allowed. Find minimal $N$ such th... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 4774,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $| \\overrightarrow{a}|=3 \\sqrt {2}$, $| \\overrightarrow{b}|=4$, $\\overrightarrow{m}= \\overrightarrow{a}+ \\overrightarrow{b}$, $\\overrightarrow{n}= \\overrightarrow{a}+λ \\overrightarrow{b}$, $ < \\overrightarrow{a}, \\overrightarrow{b} > =135^{\\circ}$, if $\\... | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{2}"
} | {
"split": "train",
"index": 4775,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any positive integers \\( m \\) and \\( n \\), define \\( r(m, n) \\) as the remainder of \\( m \\div n \\) (for example, \\( r(8,3) \\) represents the remainder of \\( 8 \\div 3 \\), so \\( r(8,3)=2 \\)). What is the smallest positive integer solution satisfying the equati... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 4776,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The curve given by the equation \\( y = 2^p x^2 + 5px - 2^{p^2} \\) intersects the \\( Ox \\) axis at points \\( A \\) and \\( B \\), and the \\( Oy \\) axis at point \\( C \\). Find the sum of all values of the parameter \\( p \\) for which the center of the circle circumscrib... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 4777,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Right triangle $ABC$ has one leg of length 9 cm, another leg of length 12 cm, and a right angle at $A$. A square has one side on the hypotenuse of triangle $ABC$ and a vertex on each of the two legs of triangle $ABC$. What is the length of one side of the square, in cm? Express... | math | {
"style": "rule",
"ground_truth": "\\frac{45}{8}"
} | {
"split": "train",
"index": 4778,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given Lara ate $\\frac{1}{4}$ of a pie and Ryan ate $\\frac{3}{10}$ of the same pie, then Cassie ate $\\frac{2}{3}$ of the pie that was left. Calculate the fraction of the original pie that was not eaten. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{20}"
} | {
"split": "train",
"index": 4779,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the shortest distance between the lines $\\frac{x+2}{2}=\\frac{y-1}{3}=\\frac{z}{1}$ and $\\frac{x-3}{-1}=\\frac{y}{1}=\\frac{z+1}{2}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5 \\sqrt{3}}{3}"
} | {
"split": "train",
"index": 4780,
"solution": "First we find the direction of a line perpendicular to both of these lines. By taking the cross product $(2,3,1) \\times(-1,1,2)=(5,-5,5)$ we find that the plane $x-y+z+3=0$ contains the first line and is parallel to the second. Now we take a point on the second lin... |
deepscaler | [
{
"role": "user",
"content": " $A, B, C$ , and $D$ are all on a circle, and $ABCD$ is a convex quadrilateral. If $AB = 13$ , $BC = 13$ , $CD = 37$ , and $AD = 47$ , what is the area of $ABCD$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "504"
} | {
"split": "train",
"index": 4781,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The exact amount of fencing that enclosed the four congruent equilateral triangular corrals shown here is reused to form one large equilateral triangular corral. What is the ratio of the total area of the four small corrals to the area of the new large corral? Express your answ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 4782,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $y=a+\\frac{b}{x}$, where $a$ and $b$ are constants, and if $y=1$ when $x=-1$, and $y=5$ when $x=-5$, then $a+b$ equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 4783,
"solution": "1. **Substitute the given values into the equation:** \n Given the equation $y = a + \\frac{b}{x}$, we substitute the values of $y$ and $x$ for the two conditions provided:\n - When $x = -1$ and $y = 1$, we have:\n \\[\n 1 = a + \\frac{b}{-1} \\implies ... |
deepscaler | [
{
"role": "user",
"content": "As shown in the diagram, squares \\( a \\), \\( b \\), \\( c \\), \\( d \\), and \\( e \\) are used to form a rectangle that is 30 cm long and 22 cm wide. Find the area of square \\( e \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 4784,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nLet \\( N \\) be the total number of students in the school before the New Year, among which \\( M \\) are boys, making up \\( k \\) percent of the total. This means \\( M = \\frac{k}{100} N \\), or \\( 100M = kN \\).\n\nAfter the New Year, the number of boys became \\( M+1 \... | math | {
"style": "rule",
"ground_truth": "197"
} | {
"split": "train",
"index": 4785,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers \\( x \\) and \\( y \\) are such that the equalities \\( \\operatorname{ctg} x - \\operatorname{ctg} y = 2 \\) and \\( 5 \\sin (2x - 2y) = \\sin 2x \\sin 2y \\) hold. Find \\( \\operatorname{tg} x \\operatorname{tg} y \\). Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "-\\frac{6}{5}"
} | {
"split": "train",
"index": 4786,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the probability that a point $P$ randomly selected within the plane region $N$ defined by the system of inequalities $\\begin{cases} 0 \\leq x \\leq 1 \\\\ 0 \\leq y \\leq e \\end{cases}$ also lies within the plane region $M$ defined by the system of inequalities $\\b... | math | {
"style": "rule",
"ground_truth": "1 - \\frac{3}{2e}"
} | {
"split": "train",
"index": 4787,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$(1)$ Solve the fractional equation: $\\frac{x}{x+1}=\\frac{2x}{3x+3}+1$;<br/>$(2)$ Simplify first ($\\frac{x+2}{{x}^{2}-2x}-\\frac{x-1}{{x}^{2}-4x+4})\\div \\frac{x+2}{{x}^{3}-4x}$, then choose a suitable number from $2$, $0$, $-1$ to substitute into the simplified result for ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"split": "train",
"index": 4788,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest four-digit negative integer congruent to $1 \\pmod{17}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1002"
} | {
"split": "train",
"index": 4789,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\tan(-405^\\circ)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 4790,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( 5 \\sin 2 \\alpha = \\sin 2^\\circ \\), find the value of \\( \\frac{\\tan (\\alpha + 1^\\circ)}{\\tan (\\alpha - 1^\\circ)} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{2}"
} | {
"split": "train",
"index": 4791,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each brick in the pyramid contains one number. Whenever possible, the number in each brick is the least common multiple of the numbers of the two bricks directly above it.\n\nWhat number could be in the bottom brick? Determine all possible options.\n\n(Hint: What is the least c... | math | {
"style": "rule",
"ground_truth": "2730"
} | {
"split": "train",
"index": 4792,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a triangle. There exists a positive real number $k$, such that if the altitudes of triangle $ABC$ are extended past $A$, $B$, and $C$, to $A'$, $B'$, and $C'$, as shown, such that $AA' = kBC$, $BB' = kAC$, and $CC' = kAB$, then triangle $A'B'C'$ is equilateral.\n\... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{\\sqrt{3}}"
} | {
"split": "train",
"index": 4793,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the following matrix $$ \\begin{pmatrix}\n11& 17 & 25& 19& 16\n24 &10 &13 & 15&3\n12 &5 &14& 2&18\n23 &4 &1 &8 &22 \n6&20&7 &21&9\n\\end{pmatrix}, $$ choose five of these elements, no two from the same row or column, in such a way that the minimum of these elements is a... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 4794,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $m$ is a positive integer, and given that $\\mathop{\\text{lcm}}[40,m]=120$ and $\\mathop{\\text{lcm}}[m,45]=180$, what is $m$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 4795,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a table, there are 30 coins: 23 ten-ruble coins and 7 five-ruble coins. Out of these, 20 coins are heads up, and the remaining 10 are tails up. What is the smallest number \\( k \\) such that among any randomly chosen \\( k \\) coins, there will always be at least one ten-ru... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 4796,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $n$: $5^{2n + 1} = \\frac{1}{25}$. 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{3}{2}"
} | {
"split": "train",
"index": 4797,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a regular hexagonal pyramid \\( M A B C D E F \\). Point \\( K \\) bisects edge \\( B M \\). Find the ratio in which the plane \\( F E K \\) divides edge \\( A M \\) (at point \\( X \\)). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2:1"
} | {
"split": "train",
"index": 4798,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In right triangle $\\triangle ABC$ with $\\angle BAC = 90^\\circ$, medians $\\overline{AD}$ and $\\overline{BE}$ are given such that $AD = 18$ and $BE = 24$. If $\\overline{AD}$ is the altitude from $A$, find the area of $\\triangle ABC$. Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "432"
} | {
"split": "train",
"index": 4799,
"solution": ""
} |
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