IMO Shortlist 1992 problem 21


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2. travnja 2012.
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For each positive integer \,n,\;S(n)\, is defined to be the greatest integer such that, for every positive integer \,k\leq S(n),\;n^{2}\, can be written as the sum of \,k\, positive squares.

a.) Prove that \,S(n)\leq n^{2}-14\, for each \,n\geq 4.
b.) Find an integer \,n\, such that \,S(n)=n^{2}-14.
c.) Prove that there are infintely many integers \,n\, such that S(n)=n^{2}-14.
Izvor: Međunarodna matematička olimpijada, shortlist 1992