data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "For positive integers $n$, let $f(n)$ return the smallest positive integer $k$ such that $\\frac{1}{k}$ has exactly $n$ digits after the decimal point. How many positive integer divisors does $f(2010)$ have?"
}
] | math | {
"style": "rule",
"ground_truth": "2011"
} | {
"index": 11900,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For positive integers $n$, let $h(n)$ return the smallest positive integer $k$ such that $\\frac{1}{k}$ has exactly $n$ digits after the decimal point, and $k$ is divisible by 3. How many positive integer divisors does $h(2010)$ have?"
}
] | math | {
"style": "rule",
"ground_truth": "4022"
} | {
"index": 11901,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For positive integers $n$, let the numbers $c(n)$ be determined by the rules $c(1) = 1$, $c(2n) = c(n)$, and $c(2n+1) = (-1)^n c(n)$. Find the value of \\[ \\sum_{n=1}^{2013} c(n) c(n+2). \\]"
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 11902,
"solution": "Note that \\begin{align*} c(2k+1)c(2k+3) &= (-1)^k c(k) (-1)^{k+1} c(k+1) \\\\ &= -c(k)c(k+1) \\\\ &= -c(2k)c(2k+2). \\end{align*} It follows that $\\sum_{n=2}^{2013} c(n)c(n+2) = \\sum_{k=1}^{1006} (c(2k)c(2k+2)+c(2k+1)c(2k+3)) = 0$, and so the desired sum is $c(1)c(3) = -1$."
} |
DeepScaleR | [
{
"role": "user",
"content": "For positive integers $n$, the number of pairs of different adjacent digits in the binary (base two) representation of $n$ can be denoted as $D(n)$. Determine the number of positive integers less than or equal to 97 for which $D(n) = 2$."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 11903,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For positive integers $n,$ let $\\tau (n)$ denote the number of positive integer divisors of $n,$ including 1 and $n.$ Define $S(n)$ by $S(n)=\\tau(1)+ \\tau(2) + \\cdots + \\tau(n).$ Let $a$ denote the number of positive integers $n \\leq 3000$ with $S(n)$ odd, and let $b$ den... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"index": 11904,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For positive integers $n,$ let $\\tau (n)$ denote the number of positive integer divisors of $n,$ including 1 and $n.$ Define $S(n)$ by $S(n)=\\tau(1)+ \\tau(2) + \\cdots + \\tau(n).$ Let $c$ denote the number of positive integers $n \\leq 1000$ with $S(n)$ odd, and let $d$ den... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 11905,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For positive integers $n,$ let $\\tau (n)$ denote the number of positive integer divisors of $n,$ including 1 and $n.$ For example, $\\tau (1)=1$ and $\\tau(6) =4.$ Define $S(n)$ by $S(n)=\\tau(1)+ \\tau(2) + \\cdots + \\tau(n).$ Let $a$ denote the number of positive integers $... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 11906,
"solution": "Let $\\Delta n$ denote the sum $1+2+3+ \\dots +n-1+n$. We can easily see from the fact \"It is well-known that $\\tau(n)$ is odd if and only if $n$ is a perfect square.\", that\n$a = (2^2-1^2) + (4^2-3^2) \\dots (44^2 - 43^2) = (2+1)(2-1)+(4+3)(4-3) \\dots (44+43)(44-43) = 1+2+3...44 ... |
DeepScaleR | [
{
"role": "user",
"content": "For positive integers $x$, let $g(x)$ be the number of blocks of consecutive 1's in the binary expansion of $x$. For example, $g(19)=2$ because $19=10011_{2}$ has a block of one 1 at the beginning and a block of two 1's at the end, and $g(7)=1$ because $7=111_{2}$ only has a si... | math | {
"style": "rule",
"ground_truth": "577"
} | {
"index": 11907,
"solution": "Solution 1. We prove that $g(1)+g(2)+\\cdots+g\\left(2^{n}\\right)=1+2^{n-2}(n+1)$ for all $n \\geq 1$, giving an answer of $1+2^{6} \\cdot 9=577$. First note that $g\\left(2^{n}\\right)=1$, and that we can view $0,1, \\ldots, 2^{n}-1$ as $n$-digit binary sequences by appending leadin... |
DeepScaleR | [
{
"role": "user",
"content": "For positive integers \\( n \\), let \\( g(n) \\) return the smallest positive integer \\( k \\) such that \\( \\frac{1}{k} \\) has exactly \\( n \\) digits after the decimal point in base 6 notation. Determine the number of positive integer divisors of \\( g(2023) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "4096576"
} | {
"index": 11908,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For positive numbers $x$ and $y$ the operation $\\spadesuit (x,y)$ is defined as \n\\[\\spadesuit (x,y) = x-\\dfrac{1}{y}\\]\nWhat is $\\spadesuit (2,\\spadesuit (2,2))$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"index": 11909,
"solution": "1. **Understanding the operation $\\spadesuit$:** The operation $\\spadesuit(x, y)$ is defined as $x - \\frac{1}{y}$. \n\n2. **Calculate $\\spadesuit(2, 2)$:**\n \\[\n \\spadesuit(2, 2) = 2 - \\frac{1}{2} = 2 - 0.5 = 1.5\n \\]\n\n3. **Substitute $\\spadesuit(2, 2)$ back into the... |
DeepScaleR | [
{
"role": "user",
"content": "For positive real numbers $a,$ $b,$ $c,$ and $d,$ compute the maximum value of\n\\[\\frac{abcd(a + b + c + d)}{(a + b)^2 (c + d)^2}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 11910,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For positive real numbers $a,$ $b,$ and $c,$ compute the maximum value of\n\\[\\frac{abc(a + b + c)}{(a + b)^2 (b + c)^2}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 11911,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For positive real numbers $a,$ $b,$ and $c,$ compute the maximum value of:\n\\[\\frac{abc(a + b + c)}{(a + b)^3 (b + c)^3}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"index": 11912,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For positive real numbers $s$, let $\\tau(s)$ denote the set of all obtuse triangles that have area $s$ and two sides with lengths $4$ and $10$. The set of all $s$ for which $\\tau(s)$ is nonempty, but all triangles in $\\tau(s)$ are congruent, is an interval $[a,b)$. Find $a^2... | math | {
"style": "rule",
"ground_truth": "736"
} | {
"index": 11913,
"solution": "Note: Archimedes15 Solution which I added an answer here are two cases. Either the $4$ and $10$ are around an obtuse angle or the $4$ and $10$ are around an acute triangle. If they are around the obtuse angle, the area of that triangle is $<20$ as we have $\\frac{1}{2} \\cdot 40 \\cdo... |
DeepScaleR | [
{
"role": "user",
"content": "For positive real numbers $x,$ $y,$ and $z,$ compute the maximum value of\n\\[\\frac{xyz(x + y + z)}{(x + y)^2 (y + z)^2}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 11914,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For positive real numbers \\(a\\), \\(b\\), and \\(c\\), compute the maximum value of\n\\[\n\\frac{abc(a + b + c + ab)}{(a + b)^3 (b + c)^3}.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"index": 11915,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For positive reals $p$ and $q$, define the remainder when $p$ is divided by $q$ as the smallest nonnegative real $r$ such that $\\frac{p-r}{q}$ is an integer. For an ordered pair $(a, b)$ of positive integers, let $r_{1}$ and $r_{2}$ be the remainder when $a \\sqrt{2}+b \\sqrt{... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 11916,
"solution": "The remainder when we divide $a \\sqrt{2}+b \\sqrt{3}$ by $\\sqrt{2}$ is defined to be the smallest non-negative real $r_{1}$ such that $\\frac{a \\sqrt{2}+b \\sqrt{3}-r_{1}}{\\sqrt{2}}$ is integral. As $\\frac{x}{\\sqrt{2}}$ is integral iff $x$ is an integral multiple of $\\sqrt{2}$,... |
DeepScaleR | [
{
"role": "user",
"content": "For rational numbers $a$ and $b$, define the operation \"$\\otimes$\" as $a \\otimes b = ab - a - b - 2$.\n(1) Calculate the value of $(-2) \\otimes 3$;\n(2) Compare the size of $4 \\otimes (-2)$ and $(-2) \\otimes 4$."
}
] | math | {
"style": "rule",
"ground_truth": "-12"
} | {
"index": 11917,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For rational numbers $x$, $y$, $a$, $t$, if $|x-a|+|y-a|=t$, then $x$ and $y$ are said to have a \"beautiful association number\" of $t$ with respect to $a$. For example, $|2-1|+|3-1|=3$, then the \"beautiful association number\" of $2$ and $3$ with respect to $1$ is $3$. <br/>... | math | {
"style": "rule",
"ground_truth": "2001000"
} | {
"index": 11918,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For real number \\( x \\), let \\( [x] \\) denote the greatest integer less than or equal to \\( x \\). Find the positive integer \\( n \\) such that \\(\\left[\\log _{2} 1\\right] + \\left[\\log _{2} 2\\right] + \\left[\\log _{2} 3\\right] + \\cdots + \\left[\\log _{2} n\\righ... | math | {
"style": "rule",
"ground_truth": "312"
} | {
"index": 11919,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers $a$ and $b$ , define $$ f(a,b) = \\sqrt{a^2+b^2+26a+86b+2018}. $$ Find the smallest possible value of the expression $$ f(a, b) + f (a,-b) + f(-a, b) + f (-a, -b). $$"
}
] | math | {
"style": "rule",
"ground_truth": "4 \\sqrt{2018}"
} | {
"index": 11920,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers $a, b$ , and $c$ the polynomial $p(x) = 3x^7 - 291x^6 + ax^5 + bx^4 + cx^2 + 134x - 2$ has $7$ real roots whose sum is $97$ . Find the sum of the reciprocals of those $7$ roots."
}
] | math | {
"style": "rule",
"ground_truth": "67"
} | {
"index": 11921,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers $a$ and $b$, define $a * b=(a-b)^2$. What is $(x-y)^2*(y-x)^2$?"
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 11922,
"solution": "1. **Understanding the operation**: Given the operation $a * b = (a-b)^2$, we need to evaluate $(x-y)^2 * (y-x)^2$.\n\n2. **Symmetry in Squaring**: Note that squaring a real number results in the same value whether the number is positive or negative. Therefore, $(-a)^2 = a^2$. Applyin... |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers $a$ and $b$, define $a \\diamond b = \\sqrt{a^2 + b^2}$. What is the value of $(5 \\diamond 12) \\diamond ((-12) \\diamond (-5))$?"
}
] | math | {
"style": "rule",
"ground_truth": "13\\sqrt{2}"
} | {
"index": 11923,
"solution": "1. **Calculate $5 \\diamond 12$:**\n \\[\n 5 \\diamond 12 = \\sqrt{5^2 + 12^2} = \\sqrt{25 + 144} = \\sqrt{169} = 13\n \\]\n\n2. **Calculate $(-12) \\diamond (-5)$:**\n \\[\n (-12) \\diamond (-5) = \\sqrt{(-12)^2 + (-5)^2} = \\sqrt{144 + 25} = \\sqrt{169} = 13\n \\]\n\n3. ... |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers $a$ and $b$, define $a\\textdollar b = (a - b)^2$. What is $(x - y)^2\\textdollar(y - x)^2$?"
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 11924,
"solution": "Given the operation $a \\textdollar b = (a - b)^2$, we need to evaluate $(x - y)^2 \\textdollar (y - x)^2$.\n\n1. **Substitute into the operation definition**:\n \\[\n (x - y)^2 \\textdollar (y - x)^2 = \\left((x - y)^2 - (y - x)^2\\right)^2\n \\]\n\n2. **Simplify the expression... |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers $a,$ $b,$ and $c,$ and a real scalar $\\lambda,$ consider the matrix\n\\[\\begin{pmatrix} a + \\lambda & b & c \\\\ b & c + \\lambda & a \\\\ c & a & b + \\lambda \\end{pmatrix}.\\] Determine all possible values of \\[\\frac{a}{b + c} + \\frac{b}{a + c} + \\fra... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 11925,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers $a,$ $b,$ and $c,$ the matrix\n\\[\\begin{pmatrix} a & b & c \\\\ b & c & a \\\\ c & a & b \\end{pmatrix}\\]is not invertible. List all possible values of\n\\[\\frac{a}{b + c} + \\frac{b}{a + c} + \\frac{c}{a + b}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 11926,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers $s,$ the intersection points of the lines $2x + 3y = 8s + 5$ and $x + 2y = 3s + 2$ are plotted. Determine the slope of the line on which all these points lie."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{7}{2}"
} | {
"index": 11927,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers $s,$ the intersection points of the lines $2x - 3y = 4s + 6$ and $2x + y = 3s + 1$ are plotted. All these points lie on a particular line. Determine the slope of this line."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{13}"
} | {
"index": 11928,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers $t \\neq 0,$ the point\n\\[(x,y) = \\left( \\frac{t + 1}{t}, \\frac{t - 1}{t} \\right)\\]is plotted. All the plotted points lie on what kind of curve?\n\n(A) Line\n(B) Circle\n(C) Parabola\n(D) Ellipse\n(E) Hyperbola\n\nEnter the letter of the correct option."... | math | {
"style": "rule",
"ground_truth": "\\text{(A)}"
} | {
"index": 11929,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers $t,$ consider the point of intersection of the triplet of lines $3x - 2y = 8t - 5$, $2x + 3y = 6t + 9$, and $x + y = 2t + 1$. All the plotted points lie on a line. Find the slope of this line."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{6}"
} | {
"index": 11930,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers $t,$ the point\n\\[(x,y) = (2^t - 3, 4^t - 5 \\cdot 2^t - 1)\\]is plotted. All the plotted points lie on what kind of curve?\n\n(A) Line\n(B) Circle\n(C) Parabola\n(D) Ellipse\n(E) Hyperbola\n\nEnter the letter of the correct option."
}
] | math | {
"style": "rule",
"ground_truth": "\\text{(C)}"
} | {
"index": 11931,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers $t,$ the point\n\\[(x,y) = (\\cos^2 t, \\sin^2 t)\\]is plotted. All the plotted points lie on what kind of curve?\n\n(A) Line\n(B) Circle\n(C) Parabola\n(D) Ellipse\n(E) Hyperbola\n\nEnter the letter of the correct option."
}
] | math | {
"style": "rule",
"ground_truth": "\\text{(A)}"
} | {
"index": 11932,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers $t,$ the point\n\\[(x,y) = \\left( \\frac{1 - t^2}{1 + t^2}, \\frac{2t}{1 + t^2} \\right)\\]is plotted. All the plotted points lie on what kind of curve?\n\n(A) Line\n(B) Circle\n(C) Parabola\n(D) Ellipse\n(E) Hyperbola\n\nEnter the letter of the correct optio... | math | {
"style": "rule",
"ground_truth": "\\text{(B)}"
} | {
"index": 11933,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers $t,$ the point\n\\[(x,y) = \\left( e^t + e^{-t}, 3 (e^t - e^{-t}) \\right)\\]is plotted. All the plotted points lie on what kind of curve?\n\n(A) Line\n(B) Circle\n(C) Parabola\n(D) Ellipse\n(E) Hyperbola\n\nEnter the letter of the correct option.\n\nNote: $e$... | math | {
"style": "rule",
"ground_truth": "\\text{(E)}"
} | {
"index": 11934,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers $t,$ the point of intersection of the lines $tx - 2y - 3t = 0$ and $x - 2ty + 3 = 0$ is plotted. All the plotted points lie on what kind of curve?\n\n(A) Line\n(B) Circle\n(C) Parabola\n(D) Ellipse\n(E) Hyperbola\n\nEnter the letter of the correct option."
}... | math | {
"style": "rule",
"ground_truth": "\\text{(E)}"
} | {
"index": 11935,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers $t,$ the point of intersection of the lines $x + 2y = 7t + 3$ and $x - y = 2t - 2$ is plotted. All the plotted points lie on a line. Find the slope of this line."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{11}"
} | {
"index": 11936,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers $w$ and $z$, $\\frac{\\frac{1}{w} + \\frac{1}{z}}{\\frac{1}{w} - \\frac{1}{z}} = 2014.$ What is $\\frac{w+z}{w-z}$?"
}
] | math | {
"style": "rule",
"ground_truth": "-2014"
} | {
"index": 11937,
"solution": "1. Start with the given equation:\n \\[\n \\frac{\\frac{1}{w} + \\frac{1}{z}}{\\frac{1}{w} - \\frac{1}{z}} = 2014\n \\]\n\n2. Multiply both sides by $\\left(\\frac{1}{w} - \\frac{1}{z}\\right)$ to eliminate the denominator:\n \\[\n \\frac{1}{w} + \\frac{1}{z} = 2014 \\left(\... |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers $x > 1,$ find the minimum value of\n\\[\\frac{x + 8}{\\sqrt{x - 1}}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 11938,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers $x$ and $y$, define $x \\spadesuit y = (x+y)(x-y)$. What is $3 \\spadesuit (4 \\spadesuit 5)$?"
}
] | math | {
"style": "rule",
"ground_truth": "-72"
} | {
"index": 11939,
"solution": "1. **Define the operation $\\spadesuit$:**\n Given $x \\spadesuit y = (x+y)(x-y)$, we need to evaluate $3 \\spadesuit (4 \\spadesuit 5)$.\n\n2. **Evaluate $4 \\spadesuit 5$:**\n \\[\n 4 \\spadesuit 5 = (4+5)(4-5) = 9 \\times (-1) = -9\n \\]\n\n3. **Substitute $-9$ into the exp... |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers $x$ and $y$, define $x\\spadesuit y = (x + y)(x - y)$. What is $3\\spadesuit(4\\spadesuit 5)$?"
}
] | math | {
"style": "rule",
"ground_truth": "-72"
} | {
"index": 11940,
"solution": "To solve the problem, we first need to evaluate $4 \\spadesuit 5$ using the operation defined as $x \\spadesuit y = (x + y)(x - y)$.\n\n1. **Calculate $4 \\spadesuit 5$:**\n \\[\n 4 \\spadesuit 5 = (4 + 5)(4 - 5) = 9 \\times (-1) = -9\n \\]\n\n2. **Next, calculate $3 \\spadesuit... |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers $x$, let \n\\[P(x)=1+\\cos(x)+i\\sin(x)-\\cos(2x)-i\\sin(2x)+\\cos(3x)+i\\sin(3x)\\]\nwhere $i = \\sqrt{-1}$. For how many values of $x$ with $0\\leq x<2\\pi$ does \n\\[P(x)=0?\\]"
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 11941,
"solution": "1. **Express $P(x)$ using Euler's formula**: Euler's formula states that $e^{i\\theta} = \\cos(\\theta) + i\\sin(\\theta)$. Using this, we can rewrite $P(x)$ as:\n \\[\n P(x) = 1 + e^{ix} - e^{2ix} + e^{3ix}\n \\]\n where $e^{ix} = \\cos(x) + i\\sin(x)$, $e^{2ix} = \\cos(2x) +... |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers $x$, let \\[f(x) = \\left\\{\n\\begin{array}{cl}\nx+2 &\\text{ if }x>3, \\\\\n2x+a &\\text{ if }x\\le 3.\n\\end{array}\n\\right.\\]What must the value of $a$ be to make the piecewise function continuous (which means that its graph can be drawn without lifting y... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 11942,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers \\( x \\) and \\( y \\) such that \\( x + y = 1 \\), determine the maximum value of the expression \\( A(x, y) = x^4 y + x y^4 + x^3 y + x y^3 + x^2 y + x y^2 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{16}"
} | {
"index": 11943,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers \\( x \\) and \\( y \\) within the interval \\([0, 12]\\):\n\n$$\nxy = (12 - x)^2 (12 - y)^2\n$$\n\nWhat is the maximum value of the product \\( xy \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "81"
} | {
"index": 11944,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers \\( x \\) and \\( y \\), define the operation \\( \\star \\) as follows: \\( x \\star y = xy + 4y - 3x \\).\n\nCompute the value of the expression\n\n$$\n((\\ldots)(((2022 \\star 2021) \\star 2020) \\star 2019) \\star \\ldots) \\star 2) \\star 1\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 11945,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers \\( x \\) and \\( y \\), simplify the equation \\(\\cfrac{\\frac{1}{x} + \\frac{1}{y}}{\\frac{1}{x} + 2\\frac{1}{y}} = 4\\) and express it as \\(\\frac{x+y}{x+2y}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{11}"
} | {
"index": 11946,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers \\( x \\), \\( \\lfloor x \\rfloor \\) denotes the greatest integer less than or equal to \\( x \\). Find the largest positive integer \\( n \\) such that the following equation holds:\n\\[\n\\lfloor \\log_2 1 \\rfloor + \\lfloor \\log_2 2 \\rfloor + \\lfloor \... | math | {
"style": "rule",
"ground_truth": "312"
} | {
"index": 11947,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers \\(x, y, z\\), the matrix\n\\[\n\\begin{pmatrix} \nx & y & z \\\\ \ny & z & x \\\\ \nz & x & y \n\\end{pmatrix}\n\\]\nis not invertible. Find all possible values of\n\\[\n\\frac{x}{y + z} + \\frac{y}{x + z} + \\frac{z}{x + y}.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 11948,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For real numbers \\(x\\), \\(y\\), and \\(z\\), consider the matrix\n\\[\n\\begin{pmatrix} x+y & x & y \\\\ x & y+z & y \\\\ y & x & x+z \\end{pmatrix}\n\\]\nDetermine whether this matrix is invertible. If not, list all possible values of\n\\[\n\\frac{x}{y + z} + \\frac{y}{x + ... | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 11949,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For some complex number $\\omega$ with $|\\omega| = 2016$ , there is some real $\\lambda>1$ such that $\\omega, \\omega^{2},$ and $\\lambda \\omega$ form an equilateral triangle in the complex plane. Then, $\\lambda$ can be written in the form $\\tfrac{a + \\sqrt{b}... | math | {
"style": "rule",
"ground_truth": "4032"
} | {
"index": 11950,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For some complex number $\\omega$ with $|\\omega| = 2,$ there is some real $\\lambda > 1$ such that $\\omega,$ $\\omega^2,$ and $\\lambda \\omega$ form an equilateral triangle in the complex plane. Find $\\lambda.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1 + \\sqrt{13}}{2}"
} | {
"index": 11951,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For some complex number $z$ with $|z| = 3,$ there is some real $\\lambda > 1$ such that $z,$ $z^2,$ and $\\lambda z$ form an equilateral triangle in the complex plane. Find $\\lambda.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1 + \\sqrt{33}}{2}"
} | {
"index": 11952,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For some constants $a$ and $b,$ let \\[f(x) = \\left\\{\n\\begin{array}{cl}\nax + b & \\text{if } x < 2, \\\\\n8 - 3x & \\text{if } x \\ge 2.\n\\end{array}\n\\right.\\]The function $f$ has the property that $f(f(x)) = x$ for all $x.$ What is $a + b?$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{3}"
} | {
"index": 11953,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For some constants $a$ and $c,$\n\\[\\begin{pmatrix} a \\\\ -1 \\\\ c \\end{pmatrix} \\times \\begin{pmatrix} 7 \\\\ 3 \\\\ 5 \\end{pmatrix} = \\begin{pmatrix} -11 \\\\ -16 \\\\ 25 \\end{pmatrix}.\\]Enter the ordered pair $(a,c).$"
}
] | math | {
"style": "rule",
"ground_truth": "(6,2)"
} | {
"index": 11954,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For some constants $x$ and $a$, the third, fourth, and fifth terms in the expansion of $(x + a)^n$ are 84, 280, and 560, respectively. Find $n.$"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 11955,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For some constants \\( c \\) and \\( d \\), let \n\\[ g(x) = \\left\\{\n\\begin{array}{cl}\ncx + d & \\text{if } x < 3, \\\\\n10 - 2x & \\text{if } x \\ge 3.\n\\end{array}\n\\right.\\]\nThe function \\( g \\) has the property that \\( g(g(x)) = x \\) for all \\( x \\). What is ... | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"index": 11956,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For some constants \\( c \\) and \\( d \\), let \\[ g(x) = \\left\\{\n\\begin{array}{cl}\ncx + d & \\text{if } x < 3, \\\\\n10 - 2x & \\text{if } x \\ge 3.\n\\end{array}\n\\right.\\] The function \\( g \\) has the property that \\( g(g(x)) = x \\) for all \\( x \\). What is \\(... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"index": 11957,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For some integer $m$, the polynomial $x^3 - 2011x + m$ has the three integer roots $a$, $b$, and $c$. Find $|a| + |b| + |c|.$"
}
] | math | {
"style": "rule",
"ground_truth": "98"
} | {
"index": 11958,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For some integer $m$, the polynomial $x^3 - 2011x + m$ has the three integer roots $a$, $b$, and $c$. Find $|a| + |b| + |c|$."
}
] | math | {
"style": "rule",
"ground_truth": "98"
} | {
"index": 11959,
"solution": "We have\n$(x-a)\\cdot (x-b)\\cdot (x-c)=x^3-(a+b+c)x+(ab+ac+bc)x-abc$ \n As a result, we have\n$a+b+c=0$\n$ab+bc+ac=-2011$\n$abc=-m$\n So, $a=-b-c$\n As a result, $ab+bc+ac=(-b-c)b+(-b-c)c+bc=-b^2-c^2-bc=-2011$\n Solve $b=\\frac {-c+\\sqrt{c^2-4(c^2-2011)}}{2}$ and $\\Delta =8044-3c^2... |
DeepScaleR | [
{
"role": "user",
"content": "For some integers $m$ and $n$, the expression $(x+m)(x+n)$ is equal to a quadratic expression in $x$ with a constant term of -12. Which of the following cannot be a value of $m$?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 11960,
"solution": "Expanding, $(x+m)(x+n)=x^{2}+n x+m x+m n=x^{2}+(m+n) x+m n$. The constant term of this quadratic expression is $m n$, and so $m n=-12$. Since $m$ and $n$ are integers, they are each divisors of -12 and thus of 12. Of the given possibilities, only 5 is not a divisor of 12, and so $m$ c... |
DeepScaleR | [
{
"role": "user",
"content": "For some integers that are not palindromes, like 91, a person can create a palindrome by repeatedly reversing the number and adding the original number to its reverse. For example, $91 + 19 = 110$. Then $110+011 = 121$, which is a palindrome, so 91 takes two steps to become a p... | math | {
"style": "rule",
"ground_truth": "176"
} | {
"index": 11961,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For some particular value of $N$, when $(a+b+c+d+1)^N$ is expanded and like terms are combined, the resulting expression contains exactly $1001$ terms that include all four variables $a, b,c,$ and $d$, each to some positive power. What is $N$?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 11962,
"solution": "1. **Identify the form of the terms**: We are interested in terms that include all four variables $a, b, c, d$ each raised to some positive power in the expansion of $(a+b+c+d+1)^N$. This means we are looking for terms of the form $a^x b^y c^z d^w 1^t$ where $x, y, z, w > 0$ and $x + ... |
DeepScaleR | [
{
"role": "user",
"content": "For some positive integer $n$ , the sum of all odd positive integers between $n^2-n$ and $n^2+n$ is a number between $9000$ and $10000$ , inclusive. Compute $n$ .\n\n*2020 CCA Math Bonanza Lightning Round #3.1*"
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 11963,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\\frac{7}{51}$ is $0.\\overline{23}_k = 0.232323..._k$. What is $k$?"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 11964,
"solution": "1. **Understanding the repeating base-$k$ representation**: Given that $0.\\overline{23}_k = 0.232323..._k$, we interpret this as the infinite series:\n \\[\n 0.232323..._k = \\frac{2}{k} + \\frac{3}{k^2} + \\frac{2}{k^3} + \\frac{3}{k^4} + \\cdots\n \\]\n\n2. **Converting the s... |
DeepScaleR | [
{
"role": "user",
"content": "For some positive integer $k$, when 60 is divided by $k^2$, the remainder is 6. What is the remainder when 100 is divided by $k$?"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 11965,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For some positive integer $n$, the number $110n^3$ has $110$ positive integer divisors, including $1$ and the number $110n^3$. How many positive integer divisors does the number $81n^4$ have?\n$\\textbf{(A) }110\\qquad\\textbf{(B) }191\\qquad\\textbf{(C) }261\\qquad\\textbf{(D)... | math | {
"style": "rule",
"ground_truth": "325"
} | {
"index": 11966,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For some positive integer $n$, the number $120n^3$ has $120$ positive integer divisors, including $1$ and the number $120n^3$. How many positive integer divisors does the number $64n^4$ have?"
}
] | math | {
"style": "rule",
"ground_truth": "375"
} | {
"index": 11967,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For some positive integer \\( n \\), the number \\( 150n^3 \\) has \\( 150 \\) positive integer divisors, including \\( 1 \\) and the number \\( 150n^3 \\). How many positive integer divisors does the number \\( 108n^5 \\) have?"
}
] | math | {
"style": "rule",
"ground_truth": "432"
} | {
"index": 11968,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For some positive integers $a$ and $b$, the product \\[\\log_a(a+1) \\cdot \\log_{a+1} (a+2) \\dotsm \\log_{b-2} (b-1) \\cdot\\log_{b-1} b\\]contains exactly $1000$ terms, and its value is $3.$ Compute $a+b.$"
}
] | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"index": 11969,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For some positive integers $a$ and $b$, the product \\[\\log_a(a+1) \\cdot \\log_{a+1} (a+2) \\dotsm \\log_{b-2} (b-1) \\cdot\\log_{b-1} b\\]contains exactly $870$ terms, and its value is $2.$ Compute $a+b.$"
}
] | math | {
"style": "rule",
"ground_truth": "930"
} | {
"index": 11970,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For some positive integers $c$ and $d$, the product \\[\\log_c(c+1) \\cdot \\log_{c+1} (c+2) \\dotsm \\log_{d-2} (d-1) \\cdot\\log_{d-1} d\\]contains exactly $930$ terms, and its value is $3.$ Compute $c+d.$"
}
] | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"index": 11971,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For some positive integers $p$, there is a quadrilateral $ABCD$ with positive integer side lengths, perimeter $p$, right angles at $B$ and $C$, $AB=2$, and $CD=AD$. How many different values of $p<2015$ are possible?\n$\\textbf{(A) }30\\qquad\\textbf{(B) }31\\qquad\\textbf{(C) ... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 11972,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For some positive real $\\alpha$, the set $S$ of positive real numbers $x$ with $\\{x\\}>\\alpha x$ consists of the union of several intervals, with total length 20.2. The value of $\\alpha$ can be expressed as $\\frac{a}{b}$, where $a, b$ are relatively prime positive integers... | math | {
"style": "rule",
"ground_truth": "4633"
} | {
"index": 11973,
"solution": "If we note that $x=\\{x\\}+\\lfloor x\\rfloor$, then we can rewrite our given inequality as $\\{x\\}>\\frac{\\alpha}{1-\\alpha}\\lfloor x\\rfloor$. However, since $\\{x\\}<1$, we know that we must have $\\frac{\\alpha}{1-\\alpha}\\lfloor x\\rfloor<\\{x\\}<1$, so each interval is of th... |
DeepScaleR | [
{
"role": "user",
"content": "For some real number $c,$ the graphs of the equation $y=|x-20|+|x+18|$ and the line $y=x+c$ intersect at exactly one point. What is $c$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 11974,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For some real number $r,$ the polynomial $8x^3 - 4x^2 - 42x + 45$ is divisible by $(x - r)^2.$ Find $r.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 11975,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For some real numbers $a$ and $b$, the equation $9x^3 + 5ax^2 + 4bx + a = 0$ has three distinct positive roots. If the sum of the base-2 logarithms of the roots is 4, what is the value of $a$?"
}
] | math | {
"style": "rule",
"ground_truth": "-144"
} | {
"index": 11976,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For testing a certain product, there are 6 different genuine items and 4 different defective items. The test continues until all defective items are identified. If all defective items are exactly identified by the 5th test, how many possible testing methods are there?"
}
] | math | {
"style": "rule",
"ground_truth": "576"
} | {
"index": 11977,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For the \"Skillful Hands\" club, Anton needs to cut several identical pieces of wire (the length of each piece is an integer number of centimeters). Initially, Anton took a piece of wire 10 meters long and was able to cut only 9 required pieces from it. Then Anton took a piece ... | math | {
"style": "rule",
"ground_truth": "111"
} | {
"index": 11978,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For the \"Skillful Hands\" club, Pavel needs to cut several identical pieces of wire (the length of each piece is an integer number of centimeters). Initially, Pavel took a wire piece of 10 meters and was able to cut only 15 pieces of the required length. Then, Pavel took a pie... | math | {
"style": "rule",
"ground_truth": "66"
} | {
"index": 11979,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For the Olympic torch relay, it is planned to select 6 cities from 8 in a certain province to establish the relay route, satisfying the following conditions. How many methods are there for each condition?\n(1) Only one of the two cities, A and B, is selected. How many methods a... | math | {
"style": "rule",
"ground_truth": "19440"
} | {
"index": 11980,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For the Shanghai World Expo, 20 volunteers were recruited, with each volunteer assigned a unique number from 1 to 20. If four individuals are to be selected randomly from this group and divided into two teams according to their numbers, with the smaller numbers in one team and ... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 11981,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For the consumer, a single discount of $n\\%$ is more advantageous than any of the following discounts:\n(1) two successive $15\\%$ discounts\n(2) three successive $10\\%$ discounts\n(3) a $25\\%$ discount followed by a $5\\%$ discount\nWhat is the smallest possible positive i... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 11982,
"solution": "Let the original price be $x$. We need to find the smallest integer $n$ such that a single discount of $n\\%$ is more advantageous than any of the given series of discounts.\n\n1. **Calculating the effective discount for option (1):**\n Two successive $15\\%$ discounts:\n \\[\n ... |
DeepScaleR | [
{
"role": "user",
"content": "For the cubic function $f(x)=ax^3+bx^2+cx+d$ ($a\\neq 0$), define: Let $f''(x)$ be the derivative of the derivative of the function $y=f(x)$, that is, the second derivative of $f(x)$. If the equation $f''(x)=0$ has a real solution $x_0$, then the point $(x_0, f(x_0))$ is called... | math | {
"style": "rule",
"ground_truth": "2012"
} | {
"index": 11983,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For the curve $ C: y = \\frac {1}{1 + x^2}$ , Let $ A(\\alpha ,\\ f(\\alpha)),\\ B\\left( - \\frac {1}{\\alpha},\\ f\\left( - \\frac {1}{\\alpha} \\right)\\right)\\ (\\alpha > 0).$ Find the minimum area bounded by the line segments $ OA,\\ OB$ and $ C,$ where $ O$ is t... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2} - \\frac{1}{2}"
} | {
"index": 11984,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For the data whose frequency histogram is shown, by how many days is the mean number of days missed per student greater than the median number of days missed per student for the 15 students? Express your answer as a common fraction.\n\n[asy]\ndraw((0,0)--(0,6),linewidth(2));\nd... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{15}\\text{ days}"
} | {
"index": 11985,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For the ellipse $16x^2 - 64x + y^2 + 4y + 4 = 0,$ find the distance between the foci."
}
] | math | {
"style": "rule",
"ground_truth": "4 \\sqrt{15}"
} | {
"index": 11986,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For the ellipse $25x^2 - 100x + 4y^2 + 8y + 16 = 0,$ find the distance between the foci."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{462}}{5}"
} | {
"index": 11987,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For the ellipse shown below, find the distance between the foci.\n\n[asy]\nunitsize(0.3 cm);\n\nint i, n = 10;\n\nfor (i = -n; i <= n; ++i) {\n draw((i,-n)--(i,n),gray(0.7));\n draw((-n,i)--(n,i),gray(0.7));\n}\n\ndraw((0,-n)--(0,n));\ndraw((-n,0)--(n,0));\n\ndraw(shift((1,1)... | math | {
"style": "rule",
"ground_truth": "8 \\sqrt{2}"
} | {
"index": 11988,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For the equation $6 x^{2}=(2 m-1) x+m+1$ with respect to $x$, there is a root $\\alpha$ satisfying the inequality $-1988 \\leqslant \\alpha \\leqslant 1988$, and making $\\frac{3}{5} \\alpha$ an integer. How many possible values are there for $m$?"
}
] | math | {
"style": "rule",
"ground_truth": "2385"
} | {
"index": 11989,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For the four-digit number $\\overline{abcd}$ (where $1 \\leq a \\leq 9$ and $0 \\leq b, c, d \\leq 9$):\n- If $a > b$, $b < c$, and $c > d$, then $\\overline{abcd}$ is called a $P$-type number;\n- If $a < b$, $b > c$, and $c < d$, then $\\overline{abcd}$ is called a $Q$-type nu... | math | {
"style": "rule",
"ground_truth": "285"
} | {
"index": 11990,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For the four-digit number \\(\\overline{abcd}\\) where \\(1 \\leqslant a \\leqslant 9\\) and \\(0 \\leqslant b, c, d \\leqslant 9\\), if \\(a > b, b < c, c > d\\), then \\(\\overline{abcd}\\) is called a \\(P\\)-type number. If \\(a < b, b > c, c < d\\), then \\(\\overline{abcd... | math | {
"style": "rule",
"ground_truth": "285"
} | {
"index": 11991,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For the fractional equation involving $x$, $\\frac{x+m}{x-2}+\\frac{1}{2-x}=3$, if it has a root with an increase, then $m=\\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 11992,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For the function $f(x) = x - 2 - \\ln x$, we know that $f(3) = 1 - \\ln 3 < 0$, $f(4) = 2 - \\ln 4 > 0$. Using the bisection method to find the approximate value of the root of $f(x)$ within the interval $(3, 4)$, we first calculate the function value $f(3.5)$. Given that $\\ln... | math | {
"style": "rule",
"ground_truth": "3.25"
} | {
"index": 11993,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For the function $f(x)= \\sqrt {2}(\\sin x+\\cos x)$, the following four propositions are given: \n$(1)$ There exists $\\alpha\\in\\left(- \\frac {\\pi}{2},0\\right)$, such that $f(\\alpha)= \\sqrt {2}$; \n$(2)$ The graph of the function $f(x)$ is symmetric about the line $x=... | math | {
"style": "rule",
"ground_truth": "(2)(3)"
} | {
"index": 11994,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For the function $f(x)=4\\sin \\left(2x+\\frac{\\pi }{3}\\right)$, the following propositions are given:\n\n$(1)$ From $f(x_1)=f(x_2)$, it can be concluded that $x_1-x_2$ is an integer multiple of $\\pi$; $(2)$ The expression for $f(x)$ can be rewritten as $f(x)=4\\cos \\left(2... | math | {
"style": "rule",
"ground_truth": "(2)(3)"
} | {
"index": 11995,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For the function $f(x)=\\sin \\left(2x+ \\frac {\\pi}{6}\\right)$, consider the following statements: \n$(1)$ The graph of the function is symmetric about the line $x=- \\frac {\\pi}{12}$; \n$(2)$ The graph of the function is symmetric about the point $\\left( \\frac {5\\... | math | {
"style": "rule",
"ground_truth": "(2)(4)"
} | {
"index": 11996,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For the function $f(x)=a- \\frac {2}{2^{x}+1}(a\\in\\mathbb{R})$\n\n$(1)$ Determine the monotonicity of the function $f(x)$ and provide a proof;\n\n$(2)$ If there exists a real number $a$ such that the function $f(x)$ is an odd function, find $a$;\n\n$(3)$ For the $a$ found in ... | math | {
"style": "rule",
"ground_truth": "\\frac {12}{5}"
} | {
"index": 11997,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For the function $y=f(x)$, if there exists $x_{0} \\in D$ such that $f(-x_{0})+f(x_{0})=0$, then the function $f(x)$ is called a \"sub-odd function\" and $x_{0}$ is called a \"sub-odd point\" of the function. Consider the following propositions: \n$(1)$ Odd functions are neces... | math | {
"style": "rule",
"ground_truth": "(1)(2)(4)(5)"
} | {
"index": 11998,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For the graph $y = mx + 3$, determine the maximum value of $a$ such that the line does not pass through any lattice points for $0 < x \\leq 50$ when $\\frac{1}{3} < m < a$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17}{51}"
} | {
"index": 11999,
"solution": ""
} |
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