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 3 of 5: The period-two family
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In plain words

If f(2) vanishes, period 2 makes all even inputs zero and all odd inputs share one arbitrary value.

f(2)=0⟹f(2t)=0, f(2t+1)=cf(2)=0\Longrightarrow f(2t)=0,\ f(2t+1)=c
Detailed analysis

The preceding step gives period 2. Since f(0)=f(2)=0f(0)=f(2)=0, every even input has value 0; by evenness every odd input has the common value c=f(1)c=f(1). Direct substitution verifies the family for every integer c.