MathLabs

Problem 4

Find all functions f:Z→Zf:\mathbb Z\to\mathbb Z such that for all integers a,b,ca,b,c with a+b+c=0a+b+c=0, 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=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).
Step 1 of 5: Normalize at zero and use symmetry
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In plain words

The zero triple first fixes the origin, and then every value is symmetric under changing the sign of its input.

f(0)=0,f(−a)=f(a)f(0)=0,\qquad f(-a)=f(a)
Detailed analysis

Putting a=b=c=0a=b=c=0 gives f(0)=0f(0)=0. Putting c=0c=0 and b=−ab=-a gives (f(a)−f(−a))2=0(f(a)-f(-a))^2=0, hence ff is even.