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rearrangement of sums, is it possible?

Source: Nordic Mathematical Contest 1989 #4

October 5, 2017
Integercombinatorics

Problem Statement

For which positive integers nn is the following statement true: if a1,a2,...,ana_1, a_2, ... , a_n are positive integers, ak≤na_k \le n for all kk and ∑k=1nak=2n\sum\limits_{k=1}^{{n}}{a_k}=2n then it is always possible to choose ai1,ai2,...,aija_{i1} , a_{i2} , ..., a_{ij} in such a way that the indices i1,i2,...,iji_1, i_2,... , i_j are different numbers, and ∑k=1jaik=n\sum\limits_{k=1}^{{{j}}}{a_{ik}}=n?
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