data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "The function $f$ satisfies \\[\nf(x) + f(2x+y) + 5xy = f(3x - y) + 2x^2 + 1\n\\]for all real numbers $x,y$. Determine the value of $f(10)$."
}
] | math | {
"style": "rule",
"ground_truth": "-49"
} | {
"index": 32500,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f$ takes nonnegative integers to real numbers, such that $f(1) = 1,$ and\n\\[f(m + n) + f(m - n) = \\frac{f(2m) + f(2n)}{2}\\]for all nonnnegative integers $m \\ge n.$ Find the sum of all possible values of $f(10).$"
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 32501,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f$, defined on the set of ordered pairs of positive integers, satisfies the following properties: \\[f(x, x) = x,\\; f(x, y) = f(y, x), {\\rm \\ and\\ } (x+y)f(x, y) = yf(x, x+y).\\] Calculate $f(14,52)$."
}
] | math | {
"style": "rule",
"ground_truth": "364"
} | {
"index": 32502,
"solution": "Since all of the function's properties contain a recursive definition except for the first one, we know that $f(x,x) = x$ in order to obtain an integer answer. So, we have to transform $f(14,52)$ to this form by exploiting the other properties. The second one doesn't help us immediate... |
DeepScaleR | [
{
"role": "user",
"content": "The function $f$, defined on the set of ordered pairs of positive integers, satisfies the following properties: \\begin{align*} f(x,x) &=x, \\\\ f(x,y) &=f(y,x), \\quad \\text{and} \\\\ (x + y) f(x,y) &= yf(x,x + y). \\end{align*}Calculate $f(14,52)$."
}
] | math | {
"style": "rule",
"ground_truth": "364"
} | {
"index": 32503,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(n)$ defined on the set of natural numbers $\\mathbf{N}$ is given by:\n$$\nf(n)=\\left\\{\\begin{array}{ll}\nn-3 & (n \\geqslant 1000); \\\\\nf[f(n+7)] & (n < 1000),\n\\end{array}\\right.\n$$\n\nWhat is the value of $f(90)$?"
}
] | math | {
"style": "rule",
"ground_truth": "999"
} | {
"index": 32504,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(n)$ is defined on the positive integers and takes non-negative integer values. $f(2)=0,f(3)>0,f(9999)=3333$ and for all $m,n:$ \\[ f(m+n)-f(m)-f(n)=0 \\text{ or } 1. \\] Determine $f(1982)$."
}
] | math | {
"style": "rule",
"ground_truth": "660"
} | {
"index": 32505,
"solution": "\nWe are given that the function \\( f(n) \\) is defined on positive integers and it takes non-negative integer values. It satisfies:\n\n\\[ f(2) = 0, \\]\n\\[ f(3) > 0, \\]\n\\[ f(9999) = 3333, \\]\n\nand for all \\( m, n \\):\n\n\\[ f(m+n) - f(m) - f(n) = 0 \\text{ or } 1. \\]\n\nWe... |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(n)$ is defined on the positive integers such that $f(f(n)) = 2n$ and $f(4n + 1) = 4n + 3$ for all positive integers $n.$ Find $f(1000).$"
}
] | math | {
"style": "rule",
"ground_truth": "1016"
} | {
"index": 32506,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x) = (m^2 - m - 1)x^m$ is a power function, and it is a decreasing function on $x \\in (0, +\\infty)$. The value of the real number $m$ is"
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 32507,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x) = x(x - m)^2$ reaches its maximum value at $x = -2$. Determine the value of $m$."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"index": 32508,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x) = x^3 + bx^2 + cx + d$ is decreasing on the interval $[-2,2]$. The maximum value of $b+c$ is __________."
}
] | math | {
"style": "rule",
"ground_truth": "-12"
} | {
"index": 32509,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x) = |\\log_3 x|$ has a range of $[0,1]$ on the interval $[a, b]$. Find the minimum value of $b - a$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 32510,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x)$ is defined by $f(x)=x^{2}-x$. What is the value of $f(4)$?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 32511,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x)$ is of the form $a x^{2}+b x+c$ for some integers $a, b$, and $c$. Given that $\\{f(177883), f(348710), f(796921), f(858522)\\} = \\{1324754875645,1782225466694,1984194627862,4388794883485\\}$ compute $a$."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 32512,
"solution": "We first match the outputs to the inputs. To start, we observe that since $a \\geq 0$ (since the answer to the problem is nonnegative), we must either have $f(858522) \\approx 4.39 \\cdot 10^{12}$ or $f(177883) \\approx 4.39 \\cdot 10^{12}$. However, since 858522 is relatively close t... |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x)$ satisfies\n\\[ f(xy) = 3xf(y) \\]\nfor all real numbers $x$ and $y$. If $f(1) = 10$, find $f(5)$."
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 32513,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x)$ satisfies\n\\[3f(x) - 2 f \\left( \\frac{1}{x} \\right) = x\\]for all $x \\neq 0.$ Find $f(4).$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"index": 32514,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x)$ satisfies\n\\[b^2 f(a) = a^2 f(b)\\]for all real numbers $a$ and $b.$ If $f(2) \\neq 0,$ find\n\\[\\frac{f(5) - f(1)}{f(2)}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 32515,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x)$ satisfies\n\\[f(2^x) + xf(2^{-x}) = 1\\]for all real numbers $x.$ Find $f(2).$"
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 32516,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x)$ satisfies\n\\[f(x + f(x)) = 4f(x)\\]for all $x,$ and $f(1) = 4.$ Find $f(21).$"
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 32517,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x)$ satisfies\n\\[f(x + y) = f(x) + f(y)\\]for all real numbers $x$ and $y,$ and $f(4) = 5.$ Find $f(5).$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{25}{4}"
} | {
"index": 32518,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x)$ satisfies\n\\[f(x + y) = f(x) f(y)\\]for all real numbers $x$ and $y.$ Find all possible values of $f(0).$\n\nEnter all the possible values, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "0,1"
} | {
"index": 32519,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x)$ satisfies\n\\[f(x + y) = f(x) f(y)\\]for all real numbers $x$ and $y.$ If $f(2) = 3,$ find $f(6).$"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 32520,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x)$ satisfies\n\\[f(x - y) = f(x) f(y)\\]for all real numbers $x$ and $y,$ and $f(x) \\neq 0$ for all real numbers $x.$ Find $f(3).$"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 32521,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x)$ satisfies\n\\[f(x) + 2f(1 - x) = 3x^2\\]for all real numbers $x.$ Find $f(3).$"
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 32522,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x)$ satisfies\n\\[f(x) + f \\left( \\frac{x + 1}{1 - 3x} \\right) = x\\]for all $x \\neq \\frac{1}{3}.$ Find $f(2).$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{48}{35}"
} | {
"index": 32523,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x)$ satisfies\n\\[f(x) - 2 f \\left( \\frac{1}{x} \\right) = 4^x\\]for all $x \\neq 0.$ Find $f(2).$"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{20}{3}"
} | {
"index": 32524,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x)$ satisfies\n\\[f(xy) = \\frac{f(x)}{y}\\]for all positive real numbers $x$ and $y.$ If $f(30) = 20,$ find $f(40).$"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 32525,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x)$ satisfies\n\\[f(xy) = f(x) f(y)\\]for all real numbers $x$ and $y,$ and $f(0) \\neq 0.$ Find $f(10).$"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 32526,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x)$ satisfies\n\\[f(xy) = xf(y)\\]for all real numbers $x$ and $y.$ If $f(1) = 25,$ find $f(79).$"
}
] | math | {
"style": "rule",
"ground_truth": "1975"
} | {
"index": 32527,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x)$ satisfies\n\\[xf(y) = yf(x)\\]for all real numbers $x$ and $y.$ If $f(15) = 20,$ find $f(3).$"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 32528,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x)$ satisfies $f(1) = 1$ and\n\\[f(x + y) = 3^y f(x) + 2^x f(y)\\]for all real numbers $x$ and $y.$ Find the function $f(x).$"
}
] | math | {
"style": "rule",
"ground_truth": "3^x - 2^x"
} | {
"index": 32529,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x)$ satisfies $f(2+x)=f(2-x)$ for all real numbers $x$. If the equation $f(x)=0$ has exactly four distinct real roots, then the sum of these roots is"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 32530,
"solution": "1. **Identify Symmetry in Function**: Given the function $f(x)$ satisfies $f(2+x) = f(2-x)$ for all real numbers $x$. This implies that $f(x)$ is symmetric about $x = 2$.\n\n2. **Roots of the Function**: We know that $f(x) = 0$ has exactly four distinct real roots. Let's denote these ... |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x)$ takes positive real numbers to real numbers, such that\n\\[xf(y) - yf(x) = f \\left( \\frac{x}{y} \\right)\\]for all positive real numbers $x$ and $y.$ Find all possible values of $f(100).$ Enter all possible values, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 32531,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x),$ defined for $0 \\le x \\le 1,$ has the following properties:\n\n(i) $f(0) = 0.$\n(ii) If $0 \\le x < y \\le 1,$ then $f(x) \\le f(y).$\n(iii) $f(1 - x) = 1 - f(x)$ for all $0 \\le x \\le 1.$\n(iv) $f \\left( \\frac{x}{3} \\right) = \\frac{f(x)}{2}$ for $0 \... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"index": 32532,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x)=(m^{2}-m-1)x^{m^{2}+m-1}$ is a power function, and it is decreasing on $(0,+\\infty)$. Find the real number $m$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 32533,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x)=(m^{2}-m-1)x^{m^{2}+m-3}$ is a power function, and when $x\\in (0,+\\infty)$, $f(x)$ is a decreasing function. Find the real number $m=$____."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 32534,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x)=1+x- \\frac {x^{2}}{2}+ \\frac {x^{3}}{3}$, $g(x)=1-x+ \\frac {x^{2}}{2}- \\frac {x^{3}}{3}$, if the function $F(x)=f(x+3)g(x-4)$, and the zeros of the function $F(x)$ are all within $[a,b]$ $(a < b$, $a$, $b\\in\\mathbb{Z})$, then the minimum value of $b-a$ ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 32535,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x)=\\frac{1}{3}x^{3}-ax^{2}+4$, and $x=2$ is a local minimum point of the function $f(x)$.\n\n(1) Find the value of the real number $a$;\n\n(2) Find the maximum and minimum values of $f(x)$ on the interval $[-1,3]$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{3}"
} | {
"index": 32536,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x)=\\log_{a}(1-x)+\\log_{a}(x+3)$ where $0 < a < 1$. \n$(1)$ Find the zeros of the function $f(x)$. \n$(2)$ If the minimum value of the function $f(x)$ is $-2$, find the value of $a$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 32537,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f: \\mathbb{Z}^{2} \\rightarrow \\mathbb{Z}$ satisfies - $f(x, 0)=f(0, y)=0$, and - $f(x, y)=f(x-1, y)+f(x, y-1)+x+y$ for all nonnegative integers $x$ and $y$. Find $f(6,12)$."
}
] | math | {
"style": "rule",
"ground_truth": "77500"
} | {
"index": 32538,
"solution": "We claim $f(x, y)=\\binom{x+y+2}{x+1}-(x+y+2)$. Indeed, the hypothesis holds true for our base cases $f(x, 0)$ and $f(0, y)$, and moreover, $f(x-1, y)+f(x, y-1)+x+y=\\binom{x+y+1}{x}+\\binom{x+y+1}{x+1}-2(x+y+1)+x+y=\\binom{x+y+2}{x+1}-(x+y+2)$. Thus, the final answer is $\\binom{20}{... |
DeepScaleR | [
{
"role": "user",
"content": "The function $f_{}^{}$ has the property that, for each real number $x,\\,$\n$f(x)+f(x-1) = x^2.\\,$\n If $f(19)=94,\\,$ what is the remainder when $f(94)\\,$ is divided by $1000$?"
}
] | math | {
"style": "rule",
"ground_truth": "561"
} | {
"index": 32539,
"solution": "\\begin{align*}f(94)&=94^2-f(93)=94^2-93^2+f(92)=94^2-93^2+92^2-f(91)=\\cdots \\\\ &= (94^2-93^2) + (92^2-91^2) +\\cdots+ (22^2-21^2)+ 20^2-f(19) \\\\ &= 94+93+\\cdots+21+400-94 \\\\ &= 4561 \\end{align*}\n So, the remainder is $\\boxed{561}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $g : \\mathbb{R} \\to \\mathbb{R}$ satisfies\n\\[g(x) + 3g(1 - x) = 2x^2 + 1\\]for all $x.$ Find $g(5).$"
}
] | math | {
"style": "rule",
"ground_truth": "-9"
} | {
"index": 32540,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $g$ is defined on positive integers as follows:\n\\[g(n) = \\left\\{\n\\begin{array}{cl}\nn + 12 & \\text{if $n < 12$}, \\\\\ng(n - 7) & \\text{if $n \\ge 12$}.\n\\end{array}\n\\right.\\]\nFind the maximum value of the function."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 32541,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $g$ is defined on the set of integers and satisfies \\[g(n)= \\begin{cases} n-4 & \\mbox{if }n\\ge 1010 \\\\ g(g(n+7)) & \\mbox{if }n<1010. \\end{cases}\\] Find $g(77)$."
}
] | math | {
"style": "rule",
"ground_truth": "1011"
} | {
"index": 32542,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $g$ is defined on the set of integers and satisfies \\[g(n)= \\begin{cases} n-5 & \\mbox{if }n\\ge 1200 \\\\ g(g(n+7)) & \\mbox{if }n<1200. \\end{cases}\\] Find $g(70)$."
}
] | math | {
"style": "rule",
"ground_truth": "1195"
} | {
"index": 32543,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $g$, defined on the set of ordered pairs of positive integers, satisfies the following properties:\n\\[\ng(x,x) = x, \\quad\ng(x,y) = g(y,x), \\quad\n(x + y) g(x,y) = yg(x, x + y).\n\\]\nCalculate $g(18,63)$."
}
] | math | {
"style": "rule",
"ground_truth": "126"
} | {
"index": 32544,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $g(x)$ satisfies\n\\[ 3g(x) + 4g(1 - x) = 6x^2 \\] for all real numbers \\(x\\). Find $g(5)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-66}{7}"
} | {
"index": 32545,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $g(x)$ satisfies\n\\[g(3^x) + 2xg(3^{-x}) = 3\\] for all real numbers $x$. Find $g(3)$."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 32546,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $g(x)$ satisfies\n\\[g(x + g(x)) = 5g(x)\\] for all \\(x\\), and \\(g(1) = 5\\). Find \\(g(26)\\)."
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"index": 32547,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $g(x)$ satisfies\n\\[g(x) - 2 g \\left( \\frac{1}{x} \\right) = 3^x\\] for all \\( x \\neq 0 \\). Find $g(2)$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{29}{9}"
} | {
"index": 32548,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $g(x)$ satisfies the equation\n\\[xg(y) = 2yg(x)\\] for all real numbers $x$ and $y$. If $g(10) = 30$, find $g(2)$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 32549,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $g(x),$ defined for $0 \\le x \\le 1,$ has the following properties:\n\n(i) $g(0) = 0.$\n(ii) If $0 \\le x < y \\le 1,$ then $g(x) \\le g(y).$\n(iii) $g(1 - x) = 1 - g(x)$ for all $0 \\le x \\le 1.$\n(iv) $g\\left( \\frac{2x}{5} \\right) = \\frac{g(x)}{3}$ for $0 \... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 32550,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $g(x),$ defined for $0 \\le x \\le 1,$ has the following properties:\n\n(i) $g(0) = 0.$\n(ii) If $0 \\le x < y \\le 1,$ then $g(x) \\le g(y).$\n(iii) $g(1 - x) = 1 - g(x)$ for all $0 \\le x \\le 1.$\n(iv) $g\\left(\\frac{x}{4}\\right) = \\frac{g(x)}{3}$ for $0 \\le... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{9}"
} | {
"index": 32551,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $x^2+px+q$ with $p$ and $q$ greater than zero has its minimum value when:"
}
] | math | {
"style": "rule",
"ground_truth": "$x=\\frac{-p}{2}$"
} | {
"index": 32552,
"solution": "1. **Identify the form of the function**: The given function is $x^2 + px + q$, which is a quadratic function in the standard form $ax^2 + bx + c$ where $a = 1$, $b = p$, and $c = q$.\n\n2. **Determine the vertex**: The vertex of a parabola represented by the quadratic function $ax^2 ... |
DeepScaleR | [
{
"role": "user",
"content": "The function $y = \\frac{2}{x}$ is defined on the interval $[1, 2]$, and its graph has endpoints $A(1, 2)$ and $B(2, 1)$. Find the linear approximation threshold of the function."
}
] | math | {
"style": "rule",
"ground_truth": "3 - 2\\sqrt{2}"
} | {
"index": 32553,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $y = x^2 + 2x - 1$ attains its minimum value on the interval $[0, 3]$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 32554,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $y= |x-1|+|2x-1|+|3x-1|+ |4x-1|+|5x-1|$ achieves its minimum value when the variable $x$ equals what value?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 32555,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $y=(m^2-m-1)x^{m^2-3m-3}$ is a power function, and it is an increasing function on the interval $(0, +\\infty)$. Find the value of $m$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 32556,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $y=2\\sin(x+\\frac{\\pi}{3})$ has an axis of symmetry at $x=\\frac{\\pi}{6}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{6}"
} | {
"index": 32557,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $y=\\frac{x^3+11x^2+38x+35}{x+3}$ can be simplified into the function $y=Ax^2+Bx+C$, defined everywhere except at $x=D$. What is the sum of the values of $A$, $B$, $C$, and $D$?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 32558,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $y=\\frac{x^3+8x^2+21x+18}{x+2}$ can be simplified into the function $y=Ax^2+Bx+C$, defined everywhere except at $x=D$. What is the sum of the values of $A$, $B$, $C$, and $D$?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 32559,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $y=f(x)$ is an even function with the smallest positive period of $4$, and when $x \\in [-2,0]$, $f(x)=2x+1$. If there exist $x\\_1$, $x\\_2$, $…x\\_n$ satisfying $0 \\leqslant x\\_1 < x\\_2 < … < x\\_n$, and $|f(x\\_1)-f(x\\_2)|+|f(x\\_2)-f(x\\_1)|+…+|f(x\\_{n-1}-... | math | {
"style": "rule",
"ground_truth": "1513"
} | {
"index": 32560,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $y=f(x)$ is an even function with the smallest positive period of 4, and when $x \\in [-2, 0]$, $f(x) = 2x + 1$. If there exist $x_1, x_2, \\ldots, x_n$ satisfying $0 \\leq x_1 < x_2 < \\ldots < x_n$, and $|f(x_1) - f(x_2)| + |f(x_2) - f(x_3)| + \\ldots + |f(x_{n-1... | math | {
"style": "rule",
"ground_truth": "1513"
} | {
"index": 32561,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $y=|x-1|+|x-2|+\\ldots+|x-10|$, when $x$ takes values in the real number range, the minimum value of $y$ is."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 32562,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function \\( f \\) satisfies the equation \\((x-1) f(x) + f\\left(\\frac{1}{x}\\right) = \\frac{1}{x-1}\\) for each value of \\( x \\) not equal to 0 and 1. Find \\( f\\left(\\frac{2016}{2017}\\right) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "2017"
} | {
"index": 32563,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function \\( f(n) \\) is an integer-valued function defined on the integers which satisfies \\( f(m + f(f(n))) = -f(f(m+1)) - n \\) for all integers \\( m \\) and \\( n \\). The polynomial \\( g(n) \\) has integer coefficients and satisfies \\( g(n) = g(f(n)) \\) for all \\... | math | {
"style": "rule",
"ground_truth": "-1992"
} | {
"index": 32564,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function \\( f(n) \\) is defined on the set of natural numbers \\( N \\) as follows:\n\\[ f(n) = \\begin{cases} \nn - 3, & \\text{ if } n \\geqslant 1000, \\\\\nf[f(n + 7)], & \\text{ if } n < 1000. \n\\end{cases} \\]\n\nWhat is the value of \\( f(90) \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "999"
} | {
"index": 32565,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function \\( f(x) = \\begin{cases} \nx + 2, & x \\leq -1, \\\\\nx^2, & -1 < x < 2, \\\\\n2x, & x \\geq 2 \n\\end{cases} \\), if \\( f(x) = 3 \\), then \\( x = \\)"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 32566,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function \\( f(x) = \\frac{x^{2}}{8} + x \\cos x + \\cos (2x) \\) (for \\( x \\in \\mathbf{R} \\)) has a minimum value of ___"
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 32567,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function \\( f(x) = \\max \\left\\{\\sin x, \\cos x, \\frac{\\sin x + \\cos x}{\\sqrt{2}}\\right\\} \\) (for \\( x \\in \\mathbb{R} \\)) has a maximum value and a minimum value. Find the sum of these maximum and minimum values."
}
] | math | {
"style": "rule",
"ground_truth": "1 - \\frac{\\sqrt{2}}{2}"
} | {
"index": 32568,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function \\( f(x) = x^{2} + a x + 3a \\) has integer roots. What is the sum of all possible values of \\( a \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 32569,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function \\( f(x) \\) has a domain of \\( \\mathbf{R} \\). For any \\( x \\in \\mathbf{R} \\) and \\( y \\neq 0 \\), \\( f(x+y)=f\\left(x y-\\frac{x}{y}\\right) \\), and \\( f(x) \\) is a periodic function. Find one of its positive periods."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1 + \\sqrt{5}}{2}"
} | {
"index": 32570,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function \\( f(x) \\) is defined on the set of real numbers, and satisfies the equations \\( f(2+x) = f(2-x) \\) and \\( f(7+x) = f(7-x) \\) for all real numbers \\( x \\). Let \\( x = 0 \\) be a root of \\( f(x) = 0 \\). Denote the number of roots of \\( f(x) = 0 \\) in th... | math | {
"style": "rule",
"ground_truth": "401"
} | {
"index": 32571,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) is continuous. For every real number \\( x \\), the equation \\( f(x) \\cdot f(f(x)) = 1 \\) holds. It is known that \\( f(1000) = 999 \\). Find \\( f(500) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{500}"
} | {
"index": 32572,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function \\( f: \\mathbf{N}^{\\star} \\rightarrow \\mathbf{R} \\) satisfies \\( f(1)=1003 \\), and for any positive integer \\( n \\), it holds that\n\\[ f(1) + f(2) + \\cdots + f(n) = n^2 f(n). \\]\n\nFind the value of \\( f(2006) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2007}"
} | {
"index": 32573,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function \\( g \\), defined on the set of ordered pairs of positive integers, satisfies the following properties:\n\\[\n\\begin{align*}\ng(x, x) &= x, \\\\\ng(x, y) &= g(y, x), \\quad \\text{and} \\\\\n(x + 2y)g(x, y) &= yg(x, x + 2y).\n\\end{align*}\n\\]\nCalculate \\( g(1... | math | {
"style": "rule",
"ground_truth": "198"
} | {
"index": 32574,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function \\( g(x) \\) satisfies\n\\[ g(x) - 2 g \\left( \\frac{1}{x} \\right) = 3^x + x \\]\nfor all \\( x \\neq 0 \\). Find \\( g(2) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-4 - \\frac{2\\sqrt{3}}{3}"
} | {
"index": 32575,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function \\( y = \\cos x + \\sin x + \\cos x \\sin x \\) has a maximum value of \\(\\quad\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2} + \\sqrt{2}"
} | {
"index": 32576,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function \\( y = \\tan(2015x) - \\tan(2016x) + \\tan(2017x) \\) has how many zeros in the interval \\([0, \\pi]\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "2017"
} | {
"index": 32577,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function \\( y = f(x+1) \\) is defined on the set of real numbers \\(\\mathbf{R}\\), and its inverse function is \\( y = f^{-1}(x+1) \\). Given that \\( f(1) = 3997 \\), find the value of \\( f(1998) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"index": 32578,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function \\(f(x) = 5x^2 - 15x - 2\\) has a minimum value when x is negative."
}
] | math | {
"style": "rule",
"ground_truth": "-13.25"
} | {
"index": 32579,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function \\[f(x) = \\left\\{ \\begin{aligned} 2x + 1 & \\quad \\text{ if } x < 3 \\\\ x^2 & \\quad \\text{ if } x \\ge 3 \\end{aligned} \\right.\\] has an inverse $f^{-1}.$ Compute the value of $f^{-1}(-3) + f^{-1}(0) + \\dots + f^{-1}(4) + f^{-1}(9).$"
}
] | math | {
"style": "rule",
"ground_truth": "3.5"
} | {
"index": 32580,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function \\[f(x) = \\left\\{ \\begin{aligned} x-2 & \\quad \\text{ if } x < 4 \\\\ \\sqrt{x} & \\quad \\text{ if } x \\ge 4 \\end{aligned} \\right.\\]has an inverse $f^{-1}.$ Find the value of $f^{-1}(-5) + f^{-1}(-4) + \\dots + f^{-1}(4) + f^{-1}(5).$"
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"index": 32581,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function \\[f(x) = \\left\\{ \\begin{aligned} x-3 & \\quad \\text{ if } x < 5 \\\\ \\sqrt{x} & \\quad \\text{ if } x \\ge 5 \\end{aligned} \\right.\\] has an inverse $f^{-1}.$ Find the value of $f^{-1}(0) + f^{-1}(1) + \\dots + f^{-1}(9).$"
}
] | math | {
"style": "rule",
"ground_truth": "291"
} | {
"index": 32582,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function \\[f(x) = \\left\\{ \\begin{aligned} x-3 & \\quad \\text{ if } x < 5 \\\\ \\sqrt{x} & \\quad \\text{ if } x \\ge 5 \\end{aligned} \\right.\\]has an inverse $f^{-1}.$ Find the value of $f^{-1}(-6) + f^{-1}(-5) + \\dots + f^{-1}(5) + f^{-1}(6).$"
}
] | math | {
"style": "rule",
"ground_truth": "94"
} | {
"index": 32583,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function defined on the set of real numbers, \\(f(x)\\), satisfies \\(f(x-1) = \\frac{1 + f(x+1)}{1 - f(x+1)}\\). Find the value of \\(f(1) \\cdot f(2) \\cdot f(3) \\cdots f(2008) + 2008\\)."
}
] | math | {
"style": "rule",
"ground_truth": "2009"
} | {
"index": 32584,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function f is defined on the set of integers and satisfies $f(n)=\\begin{cases} n-3&\\mbox{if}\\ n\\ge 1000\\\\ f(f(n+5))&\\mbox{if}\\ n<1000\\end{cases}$\n Find $f(84)$."
}
] | math | {
"style": "rule",
"ground_truth": "997"
} | {
"index": 32585,
"solution": "Assume that $f(84)$ is to be performed $n+1$ times. Then we have \\[f(84)=f^{n+1}(84)=f(f^n(84+5))\\] In order to find $f(84)$, we want to know the smallest value of \\[f^n(84+5)\\ge1000\\] Because then \\[f(84)=f(f^n(84+5))=(f^n(84+5))-3\\] From which we'll get a numerical value for ... |
DeepScaleR | [
{
"role": "user",
"content": "The function f is defined recursively by f(1)=f(2)=1 and f(n)=f(n-1)-f(n-2)+n for all integers n ≥ 3. Find the value of f(2018)."
}
] | math | {
"style": "rule",
"ground_truth": "2017"
} | {
"index": 32586,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The functions $f(x) = x^2-2x + m$ and $g(x) = x^2-2x + 4m$ are evaluated when $x = 4$. What is the value of $m$ if $2f(4) = g(4)$?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 32587,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The functions $p(x),$ $q(x),$ and $r(x)$ are all invertible. We set\n\\[f = q \\circ p \\circ r.\\]Which is the correct expression for $f^{-1}$?\n\nA. $r^{-1} \\circ q^{-1} \\circ p^{-1}$\n\nB. $p^{-1} \\circ q^{-1} \\circ r^{-1}$\n\nC. $r^{-1} \\circ p^{-1} \\circ q^{-1}$\n\n... | math | {
"style": "rule",
"ground_truth": "\\text{C}"
} | {
"index": 32588,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The game of Penta is played with teams of five players each, and there are five roles the players can play. Each of the five players chooses two of five roles they wish to play. If each player chooses their roles randomly, what is the probability that each role will have exactl... | math | {
"style": "rule",
"ground_truth": "\\frac{51}{2500}"
} | {
"index": 32589,
"solution": "Consider a graph with five vertices corresponding to the roles, and draw an edge between two vertices if a player picks both roles. Thus there are exactly 5 edges in the graph, and we want to find the probability that each vertex has degree 2. In particular, we want to find the probab... |
DeepScaleR | [
{
"role": "user",
"content": "The game of rock-scissors is played just like rock-paper-scissors, except that neither player is allowed to play paper. You play against a poorly-designed computer program that plays rock with $50 \\%$ probability and scissors with $50 \\%$ probability. If you play optimally ag... | math | {
"style": "rule",
"ground_truth": "\\frac{163}{256}"
} | {
"index": 32590,
"solution": "Since rock will always win against scissors, the optimum strategy is for you to always play rock; then, you win a game if and only if the computer plays scissors. Let $p_{n}$ be the probability that the computer plays scissors $n$ times; we want $p_{0}+p_{1}+p_{2}+p_{3}+p_{4}$. Note t... |
DeepScaleR | [
{
"role": "user",
"content": "The gardener Fedya has a miracle tree with seven branches in his garden. On each branch, there can either grow 6 apples, 5 pears, or 3 oranges. Fedya discovered that the tree has fruit of all types, with the most pears and the fewest apples.\n\nHow many fruits in total grew on ... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 32591,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The gas tank in Catherine's car is $\\frac{1}{8}$ full. When 30 litres of gas are added, the tank becomes $\\frac{3}{4}$ full. If the gas costs Catherine $\\$ 1.38$ per litre, how much will it cost her to fill the remaining quarter of the tank?"
}
] | math | {
"style": "rule",
"ground_truth": "\\$16.56"
} | {
"index": 32592,
"solution": "When Catherine adds 30 litres of gasoline, the tank goes from $\\frac{1}{8}$ full to $\\frac{3}{4}$ full. Since $\\frac{3}{4}-\\frac{1}{8}=\\frac{6}{8}-\\frac{1}{8}=\\frac{5}{8}$, then $\\frac{5}{8}$ of the capacity of the tank is 30 litres. Thus, $\\frac{1}{8}$ of the capacity of the... |
DeepScaleR | [
{
"role": "user",
"content": "The general formula of the sequence \\\\(\\{a_n\\}\\) is \\\\(a_n=n\\cos \\frac{n\\pi}{2}\\\\), and the sum of its first \\\\(n\\\\) terms is \\\\(S_n\\\\). Find \\\\(S_{2019}\\\\)."
}
] | math | {
"style": "rule",
"ground_truth": "-1010"
} | {
"index": 32593,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The geometric sequence $\\left\\{a_{n}\\right\\}$ has the first term $a_{1}=1536$, and common ratio $q=-\\frac{1}{2}$. Let $\\Pi_{n}$ denote the product of its first $n$ terms $\\left(n \\in \\mathbf{N}^{*}\\right)$. Find the value of $n$ that maximizes $\\Pi_{n}\\$."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 32594,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The geometric series $a+ar+ar^2+\\cdots$ has a sum of $12$, and the terms involving odd powers of $r$ have a sum of $5.$ What is $r$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{7}"
} | {
"index": 32595,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The germination rate of cotton seeds is $0.9$, and the probability of developing into strong seedlings is $0.6$,\n\n$(1)$ If two seeds are sown per hole, the probability of missing seedlings in this hole is _______; the probability of having no strong seedlings in this hole is ... | math | {
"style": "rule",
"ground_truth": "0.936"
} | {
"index": 32596,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The given arithmetic sequences $\\{a_{n}\\}$ and $\\{b_{n}\\}$ have respective sums of the first $n$ terms, denoted by $S_{n}$ and $T_{n}$. The ratio $\\frac{S_{n}}{T_{n}} = \\frac{3n + 1}{n + 3}$. Determine the value of $\\frac{a_{2} + a_{20}}{b_{7} + b_{15}}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{3}"
} | {
"index": 32597,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The glass gauge on a cylindrical coffee maker shows that there are $45$ cups left when the coffee maker is $36\\%$ full. How many cups of coffee does it hold when it is full?"
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"index": 32598,
"solution": "1. Let $x$ be the total capacity of the coffee maker in cups when it is full.\n2. According to the problem, when the coffee maker is $36\\%$ full, it contains $45$ cups of coffee. This can be expressed as:\n \\[\n 0.36x = 45\n \\]\n3. To find the full capacity $x$, we solve the ... |
DeepScaleR | [
{
"role": "user",
"content": "The government decided to privatize civil aviation. For each of the 127 cities in the country, the connecting airline between them is sold to one of the private airlines. Each airline must make all acquired airlines one-way but in such a way as to ensure the possibility of trav... | math | {
"style": "rule",
"ground_truth": "63"
} | {
"index": 32599,
"solution": ""
} |
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