Maths Olympiad Prep

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Algebra Difficulty 4.7 AIME Prove it United States

Problem:
Find all functions f:R→Rf: \mathbb{R} \rightarrow \mathbb{R} which satisfy
f(x+y)=f(x−y)+4xy f(x+y)=f(x-y)+4 x y
for all real numbers xx and yy.

Solution

Solution:
The answer is the functions f(x)=x2+cf(x)=x^{2}+c, where cc is a constant. It is easy to check that all such functions work, since
(x+y)2+c=(x−y)2+c+4xy (x+y)^{2}+c=(x-y)^{2}+c+4 x y
is trivially true.

Now, we prove these are the only functions. Put x=y=12ax=y=\frac{1}{2} a to obtain
f(a)=f(0)+a2 f(a)=f(0)+a^{2}
for all real numbers aa, which implies the conclusion.

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