1992 IMO Problems/Problem 2

Problem

Let $\mathbb{R}$ denote the set of all real numbers. Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that

$f\left( x^{2}+f(y) \right)= y+(f(x))^{2} \hspace{0.5cm} \forall x,y \in \mathbb{R}$

Solution

We notice that the right hand side of the equation has $y$ , therefore the only way that $f\left( x^{2}+f(y) \right)$ produces that $y$ is if $f(y)=y$ .

This makes the equation as $f\left( x^{2}+y \right)= y+(f(x))^{2}$

Since $f(y)=y$ , then $f(x)=x$ , thus $f\left( x^{2}+y \right)=x^{2}+y=y+(f(x))^{2}=y+x^{2}$ and the equation holds true.

Only solution to this problem is $f(x)=x$

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

1992 IMO (Problems) • Resources
Preceded by Problem 1	1 • 2 • 3 • 4 • 5 • 6	Followed by Problem 3
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1992 IMO Problems/Problem 2

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