I encountered into problem from book "Putnam and Beyond" which seemed to me quite interesting. However, After a long meditation I can not solve it.
Let $x_1, x_2, \dots, x_n, y_1, y_2, \dots, y_m$ be positive integers, $n,m>1$. Assume that $x_1+x_2+..+x_n=y_1+y_2+\dots+y_m<mn$. Prove that in the inequality $$x_1+x_2+..+x_n=y_1+y_2+\dots+y_m$$ one can suppress (but not all) terms in such a way that the equality is still satisfied.
I read the solution which is attached above. However, I have couple of questions.
1) WLOG they assume that $x_1\geqslant y_1$. However, I dont know how to work with the inverse inequality $x_1\leqslant y_1$. Can anyone explain how to work with it?
2) They state that $y_1$ should be moved back to the right-hand side? What if $x_1-y_1$ is already suppressed by induction hypothesis?
3) Why in the case $y_1<n$ this argument does not work? I can not even understand what namely does work.
The solution, in general, is written in unclear way. Would be grateful for clear explaining.