Olympiad problems

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I Soros Olympiad 1994-95 (Rus + Ukr), 11.5

Prove that for any natural $n>1$ there are infinitely many natural numbers $m$ such that for any nonnegative integers $k_1$,$k_2$, $...$,$k_m$, $$m \ne k_1^n+ k_2^n+... k_n^n,$$

2022 Saudi Arabia BMO + EGMO TST, 1.1

Tags: number theory , Perfect Powers , Perfect power

Find all positive integers $k$ such that the product of the first $k$ primes increased by $1$ is a power of an integer (with an exponent greater than $1$).

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