data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Given that \\( M \\) is a subset of \\(\\{1, 2, 3, \\cdots, 15\\}\\) such that the product of any 3 distinct elements of \\( M \\) is not a perfect square, determine the maximum possible number of elements in \\( M \\). Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 39049,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\cos\\alpha = \\frac{1}{7}$, $\\cos(\\alpha-\\beta) = \\frac{13}{14}$, and $0<\\beta<\\alpha<\\frac{\\pi}{2}$,\n(1) find the value of $\\tan 2\\alpha$;\n(2) determine $\\beta$. 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": 39050,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "All dwarves are either liars or knights. Liars always lie, while knights always tell the truth. Each cell of a $4 \\times 4$ board contains one dwarf. It is known that among them there are both liars and knights. Each dwarf stated: \"Among my neighbors (by edge), there are an e... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 39051,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\lfloor x \\rfloor$ denote the greatest integer less than or equal to the real number $x$, such as $\\lfloor 3.2 \\rfloor = 3$, $\\lfloor -4.5 \\rfloor = -5$. The area of the shape formed by points $(x, y)$ on the plane that satisfy $\\lfloor x \\rfloor^2 + \\lfloor y \\r... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 39052,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bug starts at a vertex of a square. On each move, it randomly selects one of the three vertices where it is not currently located, and crawls along a side of the square to that vertex. Determine the probability that the bug returns to its starting vertex on its eighth move an... | math | {
"style": "rule",
"ground_truth": "2734"
} | {
"split": "train",
"index": 39053,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify the following expression: $(x^5+x^4+x+10)-(x^5+2x^4-x^3+12).$ Express your answer as a polynomial with the degrees of the terms in decreasing order. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-x^4+x^3+x-2"
} | {
"split": "train",
"index": 39054,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An equilateral triangle lies in the Cartesian plane such that the $x$-coordinates of its vertices are pairwise distinct and all satisfy the equation $x^{3}-9 x^{2}+10 x+5=0$. Compute the side length of the triangle. Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{17}"
} | {
"split": "train",
"index": 39055,
"solution": "Let three points be $A, B$, and $C$ with $x$-coordinates $a, b$, and $c$, respectively. Let the circumcircle of $\\triangle A B C$ meet the line $y=b$ at point $P$. Then, we have $\\angle B P C=60^{\\circ} \\Longrightarrow P C=$ $\\frac{2}{\\sqrt{3}}(c-b)$. Similar... |
deepscaler | [
{
"role": "user",
"content": "Find the distance from the point $(1,2,3)$ to the line described by\n\\[\\begin{pmatrix} 6 \\\\ 7 \\\\ 7 \\end{pmatrix} + t \\begin{pmatrix} 3 \\\\ 2 \\\\ -2 \\end{pmatrix}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 39056,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( R \\) be the rectangle in the Cartesian plane with vertices at \\((0,0)\\), \\((2,0)\\), \\((2,1)\\), and \\((0,1)\\). \\( R \\) can be divided into two unit squares. Pro selects a point \\( P \\) uniformly at random in the interior of \\( R \\). Find the probability th... | math | {
"style": "rule",
"ground_truth": "3/4"
} | {
"split": "train",
"index": 39057,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $(x_n)$ is determined by the conditions: $x_0=1992,x_n=-\\frac{1992}{n} \\cdot \\sum_{k=0}^{n-1} x_k$ for $n \\geq 1$ .\nFind $\\sum_{n=0}^{1992} 2^nx_n$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1992"
} | {
"split": "train",
"index": 39058,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two similar right triangles have areas of 6 square inches and 150 square inches. The length of the hypotenuse of the smaller triangle is 5 inches. What is the sum of the lengths of the legs of the larger triangle? Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 39059,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Hooligan Vasya loves to run on the escalator in the metro, and he runs down twice as fast as he runs up. If the escalator is not working, it takes Vasya 6 minutes to run up and down. If the escalator is running downwards, it takes Vasya 13.5 minutes to run up and down. How many... | math | {
"style": "rule",
"ground_truth": "324"
} | {
"split": "train",
"index": 39060,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "find all $k$ distinct integers $a_1,a_2,...,a_k$ such that there exists an injective function $f$ from reals to themselves such that for each positive integer $n$ we have $$ \\{f^n(x)-x| x \\in \\mathbb{R} \\}=\\{a_1+n,a_2+n,...,a_k+n\\} $$ . Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "{0}"
} | {
"split": "train",
"index": 39061,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any real number $k,$ the graph of\n\\[y = 7x^2 + kx - 4k\\]passes through a fixed point $(a,b).$ Find $(a,b).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(4,112)"
} | {
"split": "train",
"index": 39062,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the coefficient of the $x^{2}$ term in the expansion of $((1-ax)(1+2x)^{4})$ is $4$, then evaluate $\\int_{\\frac{e}{2}}^{a}{\\frac{1}{x}dx}$ = \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\ln 5 - 1"
} | {
"split": "train",
"index": 39063,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\vec{m}=(2a\\cos x,\\sin x)$ and $\\vec{n}=(\\cos x,b\\cos x)$, the function $f(x)=\\vec{m}\\cdot \\vec{n}-\\frac{\\sqrt{3}}{2}$, and $f(x)$ has a y-intercept of $\\frac{\\sqrt{3}}{2}$, and the closest highest point to the y-axis has coordinates $\\left(\\frac{\... | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{6}"
} | {
"split": "train",
"index": 39064,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line passing through any two vertices of a cube has a total of 28 lines. Calculate the number of pairs of skew lines among them. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "174"
} | {
"split": "train",
"index": 39065,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Medians $\\overline{DP}$ and $\\overline{EQ}$ of $\\triangle DEF$ intersect at an angle of $60^\\circ$. If $DP = 21$ and $EQ = 27$, determine the length of side $DE$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{67}"
} | {
"split": "train",
"index": 39066,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the positive integer less than 1000 which has exactly 29 positive proper divisors. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "720"
} | {
"split": "train",
"index": 39067,
"solution": "Recall that the number $N=p_{1}^{e_{1}} p_{2}^{e_{2}} \\cdots p_{k}^{e_{k}}$ (where the $p_{i}$ are distinct primes) has exactly $(e_{1}+1)(e_{2}+1) \\cdots(e_{k}+1)$ positive integer divisors including itself. We seek $N<1000$ such that this expression is 30. Sinc... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sin \\left(2x+\\frac{\\pi }{3}\\right)+\\sin \\left(2x-\\frac{\\pi }{3}\\right)+2\\cos ^{2}(x)-1$, where $x\\in R$.\n\n(1) Simplify the function $f(x)$ in the form of $A\\sin (\\omega x+\\phi )$ $(A,\\omega > 0,0 < \\phi < \\frac{\\pi }{2})$\n\n(2) Fi... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 39068,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $a$ if the lines $2y - 2a = 6x$ and $y + 1 = (a + 6)x$ are parallel? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 39069,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Dave arrives at an airport which has twelve gates arranged in a straight line with exactly $100$ feet between adjacent gates. His departure gate is assigned at random. After waiting at that gate, Dave is told the departure gate has been changed to a different gate, again at ran... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 39070,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The probability that event $A$ occurs is $\\frac{3}{4}$; the probability that event B occurs is $\\frac{2}{3}$. \nLet $p$ be the probability that both $A$ and $B$ occur. The smallest interval necessarily containing $p$ is the interval Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "[\\frac{5}{12},\\frac{2}{3}]"
} | {
"split": "train",
"index": 39071,
"solution": "1. **Identify the given probabilities**: \n - Probability that event $A$ occurs, $P(A) = \\frac{3}{4}$.\n - Probability that event $B$ occurs, $P(B) = \\frac{2}{3}$.\n\n2. **Determine the upper bound for $p$**:\n - Since the probability that both events $A$ a... |
deepscaler | [
{
"role": "user",
"content": "The number of different arrangements of $6$ rescue teams to $3$ disaster sites, where site $A$ has at least $2$ teams and each site is assigned at least $1$ team. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "360"
} | {
"split": "train",
"index": 39072,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the maximum number of different sets consisting of three terms that form arithmetic progressions and can be chosen from a sequence of real numbers \\( a_1, a_2, \\ldots, a_{101} \\), where\n\\[ \na_1 < a_2 < a_3 < \\cdots < a_{101} .\n\\] Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "2500"
} | {
"split": "train",
"index": 39073,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Bonnie constructs a frame for a cube using 12 pieces of wire that are each eight inches long. Meanwhile, Roark uses 2-inch-long pieces of wire to create a collection of unit cube frames that are not connected. The total volume of Roark's cubes is the same as the volume of Bonni... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{128}"
} | {
"split": "train",
"index": 39074,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Star writes down the whole numbers $1$ through $40$. Emilio copies Star's numbers, but he replaces each occurrence of the digit $3$ by the digit $2$. Calculate the difference between Star's sum of the numbers and Emilio's sum of the numbers. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "104"
} | {
"split": "train",
"index": 39075,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sum of the first $n$ terms of a geometric sequence ${a_{n}}$ is $S_{n}$, if $a_{3}=4$, $S_{3}=12$, find the common ratio. 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": 39076,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $m$ be the smallest integer whose cube root is of the form $n+r$, where $n$ is a positive integer and $r$ is a positive real number less than $1/1000$. Find $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 39077,
"solution": "Note that the cube root is increasing for positive reals while its derivative is decreasing, so linear approximation gives \\[\\sqrt[3]{n^3+1} - n < \\left.\\frac{d\\sqrt[3]{x}}{dx}\\right|_{x=n^3} = \\frac{1}{3n^2}\\] and \\[\\sqrt[3]{n^3+1} - n > \\left.\\frac{d\... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$, respectively, and $(2a-c)\\cos B=b\\cos C$.\n(I) Find the measure of angle $B$;\n(II) If $\\cos A=\\frac{\\sqrt{2}}{2}$ and $a=2$, find the area of $\\triangle ABC$. Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "\\frac{3+\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 39078,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At Jefferson Summer Camp, $60\\%$ of the children play soccer, $30\\%$ of the children swim, and $40\\%$ of the soccer players swim. To the nearest whole percent, what percent of the non-swimmers play soccer? Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "51\\%"
} | {
"split": "train",
"index": 39079,
"solution": "Let's denote the total number of children at the camp as $N$. We are given the following percentages:\n- $60\\%$ of the children play soccer, which translates to $0.6N$ children.\n- $30\\%$ of the children swim, which translates to $0.3N$ children.\n- $40\\%$ of th... |
deepscaler | [
{
"role": "user",
"content": "A rectangular table with dimensions $x$ cm $\\times 80$ cm is covered with identical sheets of paper measuring 5 cm $\\times 8$ cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right of ... | math | {
"style": "rule",
"ground_truth": "77"
} | {
"split": "train",
"index": 39080,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $g : \\mathbb{R} \\to \\mathbb{R}$ be a function such that\n\\[g(g(x - y)) = g(x) g(y) - g(x) + g(y) - 2xy\\]for all $x,$ $y.$ Find the sum of all possible values of $g(1).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\sqrt{2}"
} | {
"split": "train",
"index": 39081,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the coefficient of \\( x^4 \\) in the expansion of \\( (3x + 4)^8 \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1451520"
} | {
"split": "train",
"index": 39082,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A mother purchases 5 blue plates, 2 red plates, 2 green plates, and 1 orange plate. How many ways are there for her to arrange these plates for dinner around her circular table if she doesn't want the 2 green plates to be adjacent?\n Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "588"
} | {
"split": "train",
"index": 39083,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest whole number $b$ such that 101 can be expressed in base $b$ using only two digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 39084,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x$ is a real number and $x^2-7x+6<0$, what are the possible values for $x$? Use interval notation to express your answer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(1,6)"
} | {
"split": "train",
"index": 39085,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $| \\overrightarrow{a}|=5$, $| \\overrightarrow{b}|=4$, and $\\overrightarrow{a} \\cdot \\overrightarrow{b}=-10$, find the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$ (denoted as $\\langle \\overrightarrow{a}, \\overrightarrow{b} \\rangle$). Let's t... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"split": "train",
"index": 39086,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)$ defined on the interval $[-2011, 2011]$ and satisfying $f(x_1+x_2) = f(x_1) + f(x_2) - 2011$ for any $x_1, x_2 \\in [-2011, 2011]$, and $f(x) > 2011$ when $x > 0$, determine the value of $M+N$. Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "4022"
} | {
"split": "train",
"index": 39087,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two cars, A and B, start from points A and B respectively and travel towards each other at the same time. They meet at point C after 6 hours. If car A maintains its speed and car B increases its speed by 5 km/h, they will meet 12 km away from point C. If car B maintains its spe... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 39088,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many four-digit numbers $N = \\underline{a}\\,\\underline{b}\\,\\underline{c}\\,\\underline{d}$ satisfy all of the following conditions?\n\n$4000 \\le N < 6000.$\n$N$ is a multiple of $5.$\n$3 \\le b < c \\le 6.$ Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 39089,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $x$ and $y$ satisfy the equation $x^2 + y^2 - 4x - 6y + 12 = 0$, find the minimum value of $x^2 + y^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14 - 2\\sqrt{13}"
} | {
"split": "train",
"index": 39090,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ $c$ be non-zero real numbers such that $a + b + c = 0.$ Find all possible values of\n\\[\\frac{a^3 + b^3 + c^3}{abc}.\\]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": "3"
} | {
"split": "train",
"index": 39091,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At each of the sixteen circles in the network below stands a student. A total of $3360$ coins are distributed among the sixteen students. All at once, all students give away all their coins by passing an equal number of coins to each of their neighbors in the network. After the... | math | {
"style": "rule",
"ground_truth": "280"
} | {
"split": "train",
"index": 39092,
"solution": "Since the students give the same number of gifts of coins as they receive and still end up the same number of coins, we can assume that every gift of coins has the same number of coins. Let $x$ be the number of coins in each gift of coins. There $10$ people who giv... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = (m^2 - m - 1)x^{m^2 - 2m - 1}$, if it is a power function and is an increasing function on the interval $(0, +\\infty)$, determine the real number $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 39093,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that the angles of triangle $ABC$ satisfy\n\\[\\cos 3A + \\cos 3B + \\cos 3C = 1.\\]\nTwo sides of the triangle have lengths 8 and 15. Find the maximum length of the third side assuming one of the angles is $150^\\circ$. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "\\sqrt{289 + 120\\sqrt{3}}"
} | {
"split": "train",
"index": 39094,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a rectangle \\(ABCD\\) which is cut into two parts along a dashed line, resulting in two shapes that resemble the Chinese characters \"凹\" and \"凸\". Given that \\(AD = 10\\) cm, \\(AB = 6\\) cm, and \\(EF = GH = 2\\) cm, find the total perimeter of the two shapes form... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 39095,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A box contains seven cards, each with a different integer from 1 to 7 written on it. Avani takes three cards from the box and then Niamh takes two cards, leaving two cards in the box. Avani looks at her cards and then tells Niamh \"I know the sum of the numbers on your cards is... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 39096,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $(5+7)^3+(5^3+7^3)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2196"
} | {
"split": "train",
"index": 39097,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nLet $x, y, z$ be positive numbers satisfying the following system of equations:\n\n$$\n\\left\\{\\begin{array}{l}\nx^{2} + xy + y^{2} = 12 \\\\\ny^{2} + yz + z^{2} = 9 \\\\\nz^{2} + xz + x^{2} = 21\n\\end{array}\\right.\n$$\n\nFind the value of the expression $xy + yz + xz$. ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 39098,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $a, b, c > 0$ and $(a+b)bc = 5$, find the minimum value of $2a+b+c$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"split": "train",
"index": 39099,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the domain of the function $f(x)$ is $\\mathbb{R}$, and for any $x \\in \\mathbb{Z}$, it holds that $f(x) = f(x-1) + f(x+1)$. If $f(-1) = 6$ and $f(1) = 7$, then $f(2012) + f(-2012) = \\boxed{?}$. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "-13"
} | {
"split": "train",
"index": 39100,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the remainder when \\(5x^4 - 9x^3 + 3x^2 - 7x - 30\\) is divided by \\(3x - 9\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "138"
} | {
"split": "train",
"index": 39101,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The student locker numbers at Olympic High are numbered consecutively beginning with locker number $1$. The plastic digits used to number the lockers cost two cents apiece. Thus, it costs two cents to label locker number $9$ and four cents to label locker number $10$. If it cos... | math | {
"style": "rule",
"ground_truth": "2001"
} | {
"split": "train",
"index": 39102,
"solution": "To solve this problem, we need to calculate the total cost of labeling all the lockers and match it with the given cost of $137.94. We will calculate the cost for each range of locker numbers based on the number of digits in the locker numbers.\n\n1. **Calculate th... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $\\angle A = 60^\\circ$, $AB > AC$, point $O$ is the circumcenter, and the altitudes $BE$ and $CF$ intersect at point $H$. Points $M$ and $N$ are on segments $BH$ and $HF$ respectively, such that $BM = CN$. Find the value of $\\frac{MH + NH}{OH}$. Let's thi... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 39103,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ and $y$ be positive real numbers such that\n\\[\\frac{1}{x + 2} + \\frac{1}{y + 2} = \\frac{1}{3}.\\]Find the minimum value of $x + 2y.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 + 6 \\sqrt{2}"
} | {
"split": "train",
"index": 39104,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the plane vectors $\\overrightarrow{a}$, $\\overrightarrow{b}$, $\\overrightarrow{c}$ satisfy $|\\overrightarrow{a}|=1$, $|\\overrightarrow{b}|=2$, and for all $t\\in \\mathbb{R}$, $|\\overrightarrow{b}+t\\overrightarrow{a}| \\geq |\\overrightarrow{b}-\\overrightarro... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"split": "train",
"index": 39105,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of subsets $S$ of $\\{1,2, \\ldots, 48\\}$ satisfying both of the following properties: - For each integer $1 \\leq k \\leq 24$, exactly one of $2 k-1$ and $2 k$ is in $S$. - There are exactly nine integers $1 \\leq m \\leq 47$ so that both $m$ and $m+1$ are in ... | math | {
"style": "rule",
"ground_truth": "177100"
} | {
"split": "train",
"index": 39106,
"solution": "This problem can be thought of as laying down a series of $1 \\times 2$ dominoes, with each one having either the left or right square marked. The second condition states that exactly 9 pairs of consecutive dominoes will have the leftmost one with the right square ... |
deepscaler | [
{
"role": "user",
"content": "The numbers $1, 2, \\dots, 16$ are randomly placed into the squares of a $4 \\times 4$ grid. Each square gets one number, and each of the numbers is used once. Find the probability that the sum of the numbers in each row and each column is even. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "\\frac{36}{20922789888000}"
} | {
"split": "train",
"index": 39107,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nCircles \\(\\omega_{1}\\) and \\(\\omega_{2}\\) intersect at points \\(A\\) and \\(B\\). Segment \\(PQ\\) is tangent to \\(\\omega_{1}\\) at \\(P\\) and to \\(\\omega_{2}\\) at \\(Q\\), and \\(A\\) is closer to \\(PQ\\) than \\(B\\). Point \\(X\\) is on \\(\\omega_{1}\\) such... | math | {
"style": "rule",
"ground_truth": "2 - \\sqrt{3}"
} | {
"split": "train",
"index": 39108,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The average of the data $x_1, x_2, \\ldots, x_8$ is 6, and the standard deviation is 2. Then, the average and the variance of the data $2x_1-6, 2x_2-6, \\ldots, 2x_8-6$ are Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 39109,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ivan Petrovich wants to save money for his retirement in 12 years. He decided to deposit 750,000 rubles in a bank account with an 8 percent annual interest rate. What will be the total amount in the account by the time Ivan Petrovich retires, assuming the interest is compounded... | math | {
"style": "rule",
"ground_truth": "1470000"
} | {
"split": "train",
"index": 39110,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An ordered pair $(m,n)$ of non-negative integers is called \"simple\" if the addition $m+n$ in base $10$ requires no carrying. Find the number of simple ordered pairs of non-negative integers that sum to $1492$. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 39111,
"solution": "Since no carrying over is allowed, the range of possible values of any digit of $m$ is from $0$ to the respective digit in $1492$ (the values of $n$ are then fixed). Thus, the number of ordered pairs will be $(1 + 1)(4 + 1)(9 + 1)(2 + 1) = 2\\cdot 5\\cdot 10\\cdot ... |
deepscaler | [
{
"role": "user",
"content": "Let the three sides of a triangle be integers \\( l \\), \\( m \\), and \\( n \\) with \\( l > m > n \\). It is known that \\( \\left\\{\\frac{3^l}{10^4}\\right\\} = \\left\\{\\frac{3^m}{10^4}\\right\\} = \\left\\{\\frac{3^n}{10^4}\\right\\} \\), where \\( \\{x\\} \\) denotes t... | math | {
"style": "rule",
"ground_truth": "3003"
} | {
"split": "train",
"index": 39112,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a(x+2)+b(x+2)=60$ and $a+b=12$, what is the value of $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 39113,
"solution": "The equation $a(x+2)+b(x+2)=60$ has a common factor of $x+2$ on the left side. Thus, we can re-write the equation as $(a+b)(x+2)=60$. When $a+b=12$, we obtain $12 \\cdot(x+2)=60$ and so $x+2=5$ which gives $x=3$."
} |
deepscaler | [
{
"role": "user",
"content": "When three positive integers are added in pairs, the resulting sums are 998, 1050, and 1234. What is the difference between the largest and smallest of the three original positive integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "236"
} | {
"split": "train",
"index": 39114,
"solution": "Suppose that the three integers are $x, y$ and $z$ where $x+y=998$, $x+z=1050$, and $y+z=1234$. From the first two equations, $(x+z)-(x+y)=1050-998$ or $z-y=52$. Since $z+y=1234$ and $z-y=52$, then $(z+y)+(z-y)=1234+52$ or $2z=1286$ and so $z=643$. Since $z=643$ an... |
deepscaler | [
{
"role": "user",
"content": "When four lines intersect pairwise, and no three of them intersect at the same point, find the total number of corresponding angles formed by these lines. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 39115,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular 201-sided polygon is inscribed inside a circle with center $C$. Triangles are drawn by connecting any three of the 201 vertices of the polygon. How many of these triangles have the point $C$ lying inside the triangle? Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "338350"
} | {
"split": "train",
"index": 39116,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a deck consisting of three red cards labeled $A$, $B$, $C$, three green cards labeled $A$, $B$, $C$, and three blue cards labeled $A$, $B$, $C$, calculate the probability of drawing a winning set. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{14}"
} | {
"split": "train",
"index": 39117,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of\n\\[\\frac{\\sin^6 x + \\cos^6 x + 1}{\\sin^4 x + \\cos^4 x + 1}\\]over all real values $x.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"split": "train",
"index": 39118,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An integer, whose decimal representation reads the same left to right and right to left, is called symmetrical. For example, the number 513151315 is symmetrical, while 513152315 is not. How many nine-digit symmetrical numbers exist such that adding the number 11000 to them leav... | math | {
"style": "rule",
"ground_truth": "8100"
} | {
"split": "train",
"index": 39119,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the surface area of a cube inscribed in a sphere of surface area $\\pi$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 39120,
"solution": "The sphere's radius $r$ satisfies $4 \\pi r^{2}=\\pi \\Rightarrow r=1 / 2$, so the cube has body diagonal 1 , hence side length $1 / \\sqrt{3}$. So, its surface area is $6(1 / \\sqrt{3})^{2}=2$."
} |
deepscaler | [
{
"role": "user",
"content": "Given a square region with a side length of 1 meter, and a total of 5120 beans within the square with 4009 beans within the inscribed circle, determine the approximate value of pi rounded to three decimal places. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "3.13"
} | {
"split": "train",
"index": 39121,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\( r, s, t \\) are integers, and the set \\( \\{a \\mid a = 2^r + 2^s + 2^t, 0 \\leq t < s < r\\} \\) forms a sequence \\(\\{a_n\\} \\) from smallest to largest as \\(7, 11, 13, 14, \\cdots\\), find \\( a_{36} \\). Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "131"
} | {
"split": "train",
"index": 39122,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is given that $x = -2272$ , $y = 10^3+10^2c+10b+a$ , and $z = 1$ satisfy the equation $ax + by + cz = 1$ , where $a, b, c$ are positive integers with $a < b < c$ . Find $y.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1987"
} | {
"split": "train",
"index": 39123,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $F=\\frac{6x^2+16x+3m}{6}$ be the square of an expression which is linear in $x$. Then $m$ has a particular value between: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 and 4"
} | {
"split": "train",
"index": 39124,
"solution": "1. **Expression Setup**: Given the quadratic expression $F = \\frac{6x^2 + 16x + 3m}{6}$, we simplify it to:\n \\[\n F = x^2 + \\frac{8}{3}x + \\frac{m}{2}\n \\]\n\n2. **Condition for Perfect Square**: For $F$ to be the square of a linear expression, it must ... |
deepscaler | [
{
"role": "user",
"content": "Let $A B C$ be an isosceles triangle with $A B=A C$. Let $D$ and $E$ be the midpoints of segments $A B$ and $A C$, respectively. Suppose that there exists a point $F$ on ray $\\overrightarrow{D E}$ outside of $A B C$ such that triangle $B F A$ is similar to triangle $A B C$. Co... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 39125,
"solution": "Let $\\alpha=\\angle A B C=\\angle A C B, A B=2 x$, and $B C=2 y$, so $A D=D B=A E=E C=x$ and $D E=y$. Since $\\triangle B F A \\sim \\triangle A B C$ and $B A=A C$, we in fact have $\\triangle B F A \\cong \\triangle A B C$, so $B F=B A=2 x, F A=2 y$, and $\\angle... |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle ABC$ be an isosceles triangle such that $BC = 30$ and $AB = AC.$ We have that $I$ is the incenter of $\\triangle ABC,$ and $IC = 18.$ What is the length of the inradius of the triangle? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{11}"
} | {
"split": "train",
"index": 39126,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate both the product and the sum of the least common multiple (LCM) and the greatest common divisor (GCD) of $12$ and $15$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 39127,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_1,a_2,\\cdots,a_{41}\\in\\mathbb{R},$ such that $a_{41}=a_1, \\sum_{i=1}^{40}a_i=0,$ and for any $i=1,2,\\cdots,40, |a_i-a_{i+1}|\\leq 1.$ Determine the greatest possible value of\n$(1)a_{10}+a_{20}+a_{30}+a_{40};$\n$(2)a_{10}\\cdot a_{20}+a_{30}\\cdot a_{40}.$ Let's thi... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 39128,
"solution": "\nLet \\( a_1, a_2, \\ldots, a_{41} \\in \\mathbb{R} \\) such that \\( a_{41} = a_1 \\), \\( \\sum_{i=1}^{40} a_i = 0 \\), and for any \\( i = 1, 2, \\ldots, 40 \\), \\( |a_i - a_{i+1}| \\leq 1 \\). We aim to determine the greatest possible values of:\n1. \\( a_{10... |
deepscaler | [
{
"role": "user",
"content": "Four princesses each guessed a two-digit number, and Ivan guessed a four-digit number. After they wrote their numbers in a row in some order, they got the sequence 132040530321. Find Ivan's number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5303"
} | {
"split": "train",
"index": 39129,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A projectile is launched with an initial velocity of $u$ at an angle of $\\phi$ from the horizontal. The trajectory of the projectile is given by the parametric equations:\n\\[\nx = ut \\cos \\phi,\n\\]\n\\[\ny = ut \\sin \\phi - \\frac{1}{2} gt^2,\n\\]\nwhere $t$ is time and $... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{8}"
} | {
"split": "train",
"index": 39130,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A$ be a subset of $\\{1,2,\\ldots,2020\\}$ such that the difference of any two distinct elements in $A$ is not prime. Determine the maximum number of elements in set $A$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "505"
} | {
"split": "train",
"index": 39131,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A park is in the shape of a regular hexagon $2$ km on a side. Starting at a corner, Alice walks along the perimeter of the park for a distance of $5$ km. How many kilometers is she from her starting point? Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "\\sqrt{13}"
} | {
"split": "train",
"index": 39132,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(\\mathbf{a} = \\begin{pmatrix} 2 \\\\ -1 \\\\ 3 \\end{pmatrix}\\), \\(\\mathbf{b} = \\begin{pmatrix} 4 \\\\ 5 \\\\ -2 \\end{pmatrix}\\), and \\(\\mathbf{c} = \\begin{pmatrix} 1 \\\\ 2 \\\\ 5 \\end{pmatrix}\\). Compute the following:\n\\[\n((\\mathbf{a} + \\mathbf{c}) - \... | math | {
"style": "rule",
"ground_truth": "161"
} | {
"split": "train",
"index": 39133,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many real solutions are there for $x$ in the following equation: $$(x - 5x + 12)^2 + 1 = -|x|$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 39134,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $y=\\left[x\\right]$, which is called the Gaussian function and represents the greatest integer not exceeding $x$, such as $\\left[\\pi \\right]=3$, $\\left[-2.5\\right]=-3$. The solution set of the inequality $\\frac{[x]}{[x]-4}＜0$ is ______; when $x \\gt 0$... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 39135,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Reading material: After studying square roots, Kang Kang found that some expressions containing square roots can be written as the square of another expression, such as $3+2\\sqrt{2}=({1+\\sqrt{2}})^2$. With his good thinking skills, Kang Kang made the following exploration: Le... | math | {
"style": "rule",
"ground_truth": "1+\\sqrt{5}"
} | {
"split": "train",
"index": 39136,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $(9 \\times 10^{12}) \\div (3 \\times 10^4) + (2 \\times 10^8) \\div (4 \\times 10^2)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "300,500,000"
} | {
"split": "train",
"index": 39137,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum value of the function\n\n$$\nf(x)=\\sin (x+\\sin x)+\\sin (x-\\sin x)+\\left(\\frac{\\pi}{2}-2\\right) \\sin (\\sin x)\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi - 2}{\\sqrt{2}}"
} | {
"split": "train",
"index": 39138,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Natasha and Inna each bought the same box of tea bags. It is known that one tea bag is enough for either two or three cups of tea. This box lasted Natasha for 41 cups of tea, and Inna for 58 cups of tea. How many tea bags were in the box? Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 39139,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence \\( a_1, a_2, a_3, \\ldots, a_n \\) of non-zero integers such that the sum of any 7 consecutive terms is positive and the sum of any 11 consecutive terms is negative, what is the largest possible value for \\( n \\)? Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 39140,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An infinite geometric series has common ratio $\\frac{-1}{3}$ and sum $25.$ What is the second term of the sequence? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-100}{9}"
} | {
"split": "train",
"index": 39141,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express the number $15.7$ billion in scientific notation. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1.57\\times 10^{9}"
} | {
"split": "train",
"index": 39142,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integers $a$ with $1 \\leq a \\leq 10$ is $a^{2014}+a^{2015}$ divisible by 5? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 39143,
"solution": "First, we factor $a^{2014}+a^{2015}$ as $a^{2014}(1+a)$. If $a=5$ or $a=10$, then the factor $a^{2014}$ is a multiple of 5, so the original expression is divisible by 5. If $a=4$ or $a=9$, then the factor $(1+a)$ is a multiple of 5, so the original expression is di... |
deepscaler | [
{
"role": "user",
"content": "In the sequence $\\{a_{n}\\}$, where $a_{1}=1$, $a_{n} \\gt 0$, and the sum of the first $n$ terms is $S_{n}$. If ${a_n}=\\sqrt{{S_n}}+\\sqrt{{S_{n-1}}}$ for $n \\in \\mathbb{N}^*$ and $n \\geqslant 2$, then the sum of the first $15$ terms of the sequence $\\{\\frac{1}{{{a_n}{a... | math | {
"style": "rule",
"ground_truth": "\\frac{15}{31}"
} | {
"split": "train",
"index": 39144,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, angle $C$ is a right angle and the altitude from $C$ meets $\\overline{AB}$ at $D$. The lengths of the sides of $\\triangle ABC$ are integers, $BD=29^2$, and $\\sin B = p/q$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. Let's think st... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 39145,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider pairs $(f,g)$ of functions from the set of nonnegative integers to itself such that \n[list]\n[*]$f(0) \\geq f(1) \\geq f(2) \\geq \\dots \\geq f(300) \\geq 0$\n[*]$f(0)+f(1)+f(2)+\\dots+f(300) \\leq 300$\n[*]for any 20 nonnegative integers $n_1, n_2, \\dots, n_{20}$, ... | math | {
"style": "rule",
"ground_truth": "115440"
} | {
"split": "train",
"index": 39146,
"solution": "\nConsider pairs \\((f, g)\\) of functions from the set of nonnegative integers to itself such that:\n- \\(f(0) \\geq f(1) \\geq f(2) \\geq \\dots \\geq f(300) \\geq 0\\),\n- \\(f(0) + f(1) + f(2) + \\dots + f(300) \\leq 300\\),\n- for any 20 nonnegative integers \... |
deepscaler | [
{
"role": "user",
"content": "The positive number $a$ is chosen such that the terms $20, a, \\frac{5}{4}$ are the first, second and third terms, respectively, of a geometric sequence. What is the value of $a$, if $a$ is positive? Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 39147,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a positive integer $n$ such that $(n+1)! + (n+2)! = n! \\cdot 440$. What is the sum of the digits of $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 39148,
"solution": "\nWe start with the given equation:\n\\[\n(n+1)! + (n+2)! = n! \\cdot 440\n\\]\n\nFirst, we factor out $(n+1)!$ from the left-hand side:\n\\[\n(n+1)! + (n+2)! = (n+1)! + (n+2)(n+1)! = (n+1)!\\left(1 + (n+2)\\right)\n\\]\n\nThis simplifies to:\n\\[\n(n+1)!(n+3) = 44... |
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