2019 IMO Shortlist N7

Prove that there is a constant $c>0$ and infinitely many positive integers $n$ with the following property: there are infinitely many positive integers that cannot be expressed as the sum of fewer than $c n \log (n)$ pairwise coprime $n^{\text {th }}$ powers. (Canada)

证明存在常数 $c>0$ 和无穷多个正整数 $n$，且具有以下性质：有无穷多个正整数不能表示为小于 $c n \log (n)$ 成对互质 $n^{\text {th }}$ 次幂之和。 (加拿大)