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Determine the value of $ab$ if $\log_8a+\log_4b^2=5$ and $\log_8b+\log_4a^2=7$
| 512
|
[
{
"content": "求解下述奥林匹克竞赛题，最终答案是一个整数。\n```\nDetermine the value of $ab$ if $\\log_8a+\\log_4b^2=5$ and $\\log_8b+\\log_4a^2=7$\n```\n请逐步推理，并将最终答案写在 \\boxed{} 中。\n",
"role": "user"
}
] |
Three circles, each of radius $3$ , are drawn with centers at $(14, 92)$ $(17, 76)$ , and $(19, 84)$ . A line passing through $(17,76)$ is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the absolute value of the slope of this line?
| 24
|
[
{
"content": "求解下述奥林匹克竞赛题，最终答案是一个整数。\n```\nThree circles, each of radius $3$ , are drawn with centers at $(14, 92)$ $(17, 76)$ , and $(19, 84)$ . A line passing through $(17,76)$ is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the absolute value of the slope of this line?\n```\n请逐步推理，并将最终答案写在 \\boxed{} 中。\n",
"role": "user"
}
] |
The equation $z^6+z^3+1=0$ has complex roots with argument $\theta$ between $90^\circ$ and $180^\circ$ in the complex plane . Determine the degree measure of $\theta$
| 160
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[
{
"content": "求解下述奥林匹克竞赛题，最终答案是一个整数。\n```\nThe equation $z^6+z^3+1=0$ has complex roots with argument $\\theta$ between $90^\\circ$ and $180^\\circ$ in the complex plane . Determine the degree measure of $\\theta$\n```\n请逐步推理，并将最终答案写在 \\boxed{} 中。\n",
"role": "user"
}
] |
In tetrahedron $ABCD$ edge $AB$ has length 3 cm. The area of face $ABC$ is $15\mbox{cm}^2$ and the area of face $ABD$ is $12 \mbox { cm}^2$ . These two faces meet each other at a $30^\circ$ angle. Find the volume of the tetrahedron in $\mbox{cm}^3$
| 20
|
[
{
"content": "求解下述奥林匹克竞赛题，最终答案是一个整数。\n```\nIn tetrahedron $ABCD$ edge $AB$ has length 3 cm. The area of face $ABC$ is $15\\mbox{cm}^2$ and the area of face $ABD$ is $12 \\mbox { cm}^2$ . These two faces meet each other at a $30^\\circ$ angle. Find the volume of the tetrahedron in $\\mbox{cm}^3$\n```\n请逐步推理，并将最终答案写在 \\boxed{} 中。\n",
"role": "user"
}
] |
A gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let $\frac m n$ in lowest terms be the probability that no two birch trees are next to one another. Find $m+n$
| 106
|
[
{
"content": "求解下述奥林匹克竞赛题，最终答案是一个整数。\n```\nA gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let $\\frac m n$ in lowest terms be the probability that no two birch trees are next to one another. Find $m+n$\n```\n请逐步推理，并将最终答案写在 \\boxed{} 中。\n",
"role": "user"
}
] |
function $f$ is defined for all real numbers and satisfies $f(2+x)=f(2-x)$ and $f(7+x)=f(7-x)$ for all $x$ . If $x=0$ is a root for $f(x)=0$ , what is the least number of roots $f(x)=0$ must have in the interval $-1000\leq x \leq 1000$
| 401
|
[
{
"content": "求解下述奥林匹克竞赛题，最终答案是一个整数。\n```\nfunction $f$ is defined for all real numbers and satisfies $f(2+x)=f(2-x)$ and $f(7+x)=f(7-x)$ for all $x$ . If $x=0$ is a root for $f(x)=0$ , what is the least number of roots $f(x)=0$ must have in the interval $-1000\\leq x \\leq 1000$\n```\n请逐步推理，并将最终答案写在 \\boxed{} 中。\n",
"role": "user"
}
] |
Find the value of $10\cot(\cot^{-1}3+\cot^{-1}7+\cot^{-1}13+\cot^{-1}21).$
| 15
|
[
{
"content": "求解下述奥林匹克竞赛题，最终答案是一个整数。\n```\nFind the value of $10\\cot(\\cot^{-1}3+\\cot^{-1}7+\\cot^{-1}13+\\cot^{-1}21).$\n```\n请逐步推理，并将最终答案写在 \\boxed{} 中。\n",
"role": "user"
}
] |
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
| 38
|
[
{
"content": "求解下述奥林匹克竞赛题，最终答案是一个整数。\n```\nWhat is the largest even integer that cannot be written as the sum of two odd composite numbers?\n```\n请逐步推理，并将最终答案写在 \\boxed{} 中。\n",
"role": "user"
}
] |
Let $x$ $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\log_x w = 24$ $\log_y w = 40$ and $\log_{xyz} w = 12$ . Find $\log_z w$
| 60
|
[
{
"content": "求解下述奥林匹克竞赛题，最终答案是一个整数。\n```\nLet $x$ $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\\log_x w = 24$ $\\log_y w = 40$ and $\\log_{xyz} w = 12$ . Find $\\log_z w$\n```\n请逐步推理，并将最终答案写在 \\boxed{} 中。\n",
"role": "user"
}
] |
Let $f(x)=|x-p|+|x-15|+|x-p-15|$ , where $0 < p < 15$ . Determine the minimum value taken by $f(x)$ for $x$ in the interval $p \leq x\leq15$
| 15
|
[
{
"content": "求解下述奥林匹克竞赛题，最终答案是一个整数。\n```\nLet $f(x)=|x-p|+|x-15|+|x-p-15|$ , where $0 < p < 15$ . Determine the minimum value taken by $f(x)$ for $x$ in the interval $p \\leq x\\leq15$\n```\n请逐步推理，并将最终答案写在 \\boxed{} 中。\n",
"role": "user"
}
] |
What is the product of the real roots of the equation $x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}$
| 20
|
[
{
"content": "求解下述奥林匹克竞赛题，最终答案是一个整数。\n```\nWhat is the product of the real roots of the equation $x^2 + 18x + 30 = 2 \\sqrt{x^2 + 18x + 45}$\n```\n请逐步推理，并将最终答案写在 \\boxed{} 中。\n",
"role": "user"
}
] |
Suppose that the sum of the squares of two complex numbers $x$ and $y$ is $7$ and the sum of the cubes is $10$ . What is the largest real value that $x + y$ can have?
| 4
|
[
{
"content": "求解下述奥林匹克竞赛题，最终答案是一个整数。\n```\nSuppose that the sum of the squares of two complex numbers $x$ and $y$ is $7$ and the sum of the cubes is $10$ . What is the largest real value that $x + y$ can have?\n```\n请逐步推理，并将最终答案写在 \\boxed{} 中。\n",
"role": "user"
}
] |
Let $a_n=6^{n}+8^{n}$ . Determine the remainder upon dividing $a_ {83}$ by $49$
| 35
|
[
{
"content": "求解下述奥林匹克竞赛题，最终答案是一个整数。\n```\nLet $a_n=6^{n}+8^{n}$ . Determine the remainder upon dividing $a_ {83}$ by $49$\n```\n请逐步推理，并将最终答案写在 \\boxed{} 中。\n",
"role": "user"
}
] |
Twenty five of King Arthur's knights are seated at their customary round table. Three of them are chosen - all choices being equally likely - and are sent off to slay a troublesome dragon. Let $P$ be the probability that at least two of the three had been sitting next to each other. If $P$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?
| 57
|
[
{
"content": "求解下述奥林匹克竞赛题，最终答案是一个整数。\n```\nTwenty five of King Arthur's knights are seated at their customary round table. Three of them are chosen - all choices being equally likely - and are sent off to slay a troublesome dragon. Let $P$ be the probability that at least two of the three had been sitting next to each other. If $P$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?\n```\n请逐步推理，并将最终答案写在 \\boxed{} 中。\n",
"role": "user"
}
] |
What is the largest $2$ -digit prime factor of the integer $n = {200\choose 100}$
| 61
|
[
{
"content": "求解下述奥林匹克竞赛题，最终答案是一个整数。\n```\nWhat is the largest $2$ -digit prime factor of the integer $n = {200\\choose 100}$\n```\n请逐步推理，并将最终答案写在 \\boxed{} 中。\n",
"role": "user"
}
] |
Find the minimum value of $\frac{9x^2\sin^2 x + 4}{x\sin x}$ for $0 < x < \pi$
| 12
|
[
{
"content": "求解下述奥林匹克竞赛题，最终答案是一个整数。\n```\nFind the minimum value of $\\frac{9x^2\\sin^2 x + 4}{x\\sin x}$ for $0 < x < \\pi$\n```\n请逐步推理，并将最终答案写在 \\boxed{} 中。\n",
"role": "user"
}
] |
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