2021 GCIME

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 $AB\cdot BC\cdot r$ ?

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2021 GCIME

Problem 1

Problem 2

Problem 3