plagueis wrote: Let $k$ be a positive real number. Determine all functions $f:[-k, k]\rightarrow[0, k]$ satisfying the equation $$f(x)^2+f(y)^2-2xy=k^2+f(x+y)^2$$for any $x, y\in[-k, k]$ such that $x+y\in[-k, k]$. Let $g(x)=f(x)^2+x^2$.Then $0\leq g(x)=f(x)^2+x^2\leq k^2+k^2=2k^2$.So $g$ is bounded.This turns the original assertion into $g(x)+g(y)=g(0)+g(x+y)$.Now let $h(x)=g(x)-g(0)$ then $h(x+y)=h(x)+h(y)$ so $h \; \text{bounded+additive}\implies \text{linear}$.So let $h(x)=ax+b$.Then $g(x)=ax+b+c$ where $c=g(0)$.Thus $f(x)^2=(ax+b+c)-x^2\geq 0$.So $\boxed{f(x)=(ax+b+c)-x^2}$ is a solution when $k^2\geq (ax+b+c)-x^2\geq 0$.From here it is easy to see obtain the desired bounds fo $a,b,c$