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n rare coins of denominations 1,2,.., n into 5 boxes unter 4 conditions

Source: Mexican Mathematical Olympiad 2001 OMM P6

July 30, 2018
combinatoricscoins

Problem Statement

A collector of rare coins has coins of denominations 1,2,...,n1, 2,..., n (several coins for each denomination). He wishes to put the coins into 55 boxes so that: (1) in each box there is at most one coin of each denomination; (2) each box has the same number of coins and the same denomination total; (3) any two boxes contain all the denominations; (4) no denomination is in all 55 boxes. For which nn is this possible?
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