This is a stronger version of statement from this question.

Let $n \ge 2$ be an integer. Suppose we have two tuples $(x_1, x_2, \dots, x_k), (y_1, y_2, \dots, y_m)$ of positive integers not larger than $n$. It is also known that:

  1. $\displaystyle \sum_{i\,=\,1}^k x_i > \sum_{i\,=\,1}^m y_i$;
  2. $(x_1, x_2, \dots, x_k) \ne (n, n, \dots, n)$;
  3. $m \ge n-1$.

Then there exist $p,q,r,s$ such that $1 \le p \le q \le k,\, 1 \le r \le s \le m$ and $$\sum_{i\,=\,p}^q x_i = \sum_{i\,=\,r}^s y_i.$$

Can you prove it?

Avowal. I know the proof. So let me omit usual 'justification' clause. Consider this a challenge question. I can give hints instead (only if it turns out to be hard).


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