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2021 GCIME Problems

Problem 1

Let $\pi(n)$ denote the number of primes less than or equal to $n$. Suppose $\pi(a)^{\pi(b)}=\pi(b)^{\pi(a)}=c$. For some fixed $c$ what is the maximum possible number of solutions $(a, b, c)$ but not exceeding $99$?

Solution

Problem 2

Let $N$ denote the number of solutions to the given equation: \[\lfloor\sqrt{n}\rfloor+\lfloor\sqrt[3]{n}\rfloor+\lfloor\sqrt[4]{n}\lfloor+\lfloor\sqrt[5]{n}\rfloor=100\] What is the value of $N$?

Solution


Problem 3

Let $ABCD$ be a cyclic kite. Let $r\in\mathbb{N}$ be the inradius of $ABCD$. Suppose $AB\cdot BC\cdot r$ is a perfect square. What is the smallest value of $AB\cdot BC\cdot r$?

Solution


Problem 4

Define $H(m)$ as the harmonic mean of all the divisors of $m$. Find the positive integer $n<1000$ for which $\frac{H(n)}{n}$ is the minimum amongst all $1<n\leq 1000$.

Solution


Problem 5

Let $x$ be a real number such that \[\frac{\sin^{4}x}{20}+\frac{\cos^{4}x}{21}=\frac{1}{41}\] If the value of \[\frac{\sin^{6}x}{20^{3}}+\frac{\cos^{6}x}{21^{3}}\] can be expressed as $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, then what is the remainder when $m+n$ is divided by $1000$?

Solution

Problem 6

Two scales used to measure temperature are degrees Fahrenheit ($F$) and degrees Celsius ($C$) and the two are related by the formula $F=\tfrac{9}{5}C+32$. When a two-digit integer degree temperature $n$ in Celcius is converted to Fahrenheit and rounded to the nearest integer degree, it turns out the ones and tens digits of the original Celsius temperature $n$ sometimes switch places to give the rounded Fahrenheit equivalent. Find the sum of all two-digit integer values of $n$ for which this happens.

Solution

Problem 7

Let $a_{n}$ denote the units digit of ${{(4n)^{(3n)}}^{(2n)}}^{n}$. Then find the sum of all positive integers $n<1000$ such that \[\sum_{i=1}^{n}a_{i}<4n.\]

Solution

Problem 8

A basketball club decided to label every basketball in the club. After labeling all $n$ of the balls, the labeler noticed that exactly half of the balls had the digit $1$. Find the sum of all possible three-digit integer values of $n$.

Solution

Problem 9

$\triangle ABC$ has perimeter $60$, and points $D, E,$ and $F$ are chosen on sides $BC, AC,$ and $AB$ respectively. If the circumcircles of triangles $\triangle AFE, \triangle BFD,$ and $\triangle CED$ all pass through the orthocenter of $\triangle DEF,$ then the maximum possible area of $\triangle DEF$ can be written as $a\sqrt{b}$ for squarefree $b$. What is $a+b$?

Solution

Problem 10

Let $x, y,$ and $z$ be randomly chosen real numbers from the interval $[-10, 10]$. Let the probability that these randomly chosen $x, y,$ and $z$ satisfy the following inequality \[10(|x|+|y|+|z|)\geq 100\geq x^{2}+y^{2}+z^{2}\] be $\tfrac{m\pi-n}{p}$ where $m, n,$ and $p$ are relatively prime positive integers and $m$ and $p$ are relatively prime. Find $m+n+p$.

Solution

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