olymmath

ISL 2012 A1

    Find all the functions f:Z→Zf: \mathbb{Z} \rightarrow \mathbb{Z} such that f(a)2+f(b)2+f(c)2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a)f(a)^{2}+f(b)^{2}+f(c)^{2}=2 f(a) f(b)+2 f(b) f(c)+2 f(c) f(a) for all integers a,b,ca, b, c satisfying a+b+c=0a+b+c=0.

    Official text Transcription from NuminaMath olympiads, checked word for word against the official paper. Quoted with source.

    Official text: IMO 2012 Shortlist booklet, page 4.

    Solutions

    Official solution Solution 1 by the IMO Problem Selection Committee

    The terms of the source let olymmath link to this solution but not copy it. Read the solution (IMO official site).

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