OIMU 2010 Problem 7

(a) Prove that, for any positive integers $m\le \ell$ given, there is a positive integer $n$ and positive integers $x_1,\cdots,x_n,y_1,\cdots,y_n$ such that the equality $$\sum_{i=1}^nx_i^k=\sum_{i=1}^ny_i^k$$ holds for every $k=1,2,\cdots,m-1,m+1,\cdots,\ell$ , but does not hold for $k=m$ .

(b) Prove that there is a solution of the problem, where all numbers $x_1,\cdots,x_n,y_1,\cdots,y_n$ are distinct.

Proposed by Ilya Bogdanov and Géza Kós.