Track / Stage 5 / 14 of 400 #614 of 1964

Problem 614

AIME late

Algebra Difficulty 5.0 Find the answer

Given $t$ is a real number. Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$ such that
$f(x+t+f(y))=f(f(x))+f(t)+y .$
(2014, Croatian Mathematical Olympiad)

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

Let $x=y=-t$ , we get $f(t)=t$ .
Let $x=-t, y=t$ , we get $f(f(-t))=-t$ .
Let $f(-t)=a, x=t, y=a$ , we get $a=-t$ .
Finally, let $y=-t$ and $x=-t$ , we get the solution of the function as $f(x)=x$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.