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 $99$ ?

Problem 2

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

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 $AB\cdot BC\cdot r$ ?

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 $1<n\leq 1000$ .

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 $1000$ ?

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