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Philippine Mathematical Olympiad

Algebra Difficulty 5.3 AIME, harder Prove it Philippines

Problem:
Let $f: \mathbb{R} \longrightarrow \mathbb{R}$ be a function such that $x f(y)=y f(x)$ for all $x, y \in \mathbb{R}$ . Find the intersection of the graphs of $y=f(x)$ and $y=x^{2}+1$ if $f(1)=-1$ .

Solution

Solution:
(ans $\phi=$ Null set.
We have that $\frac{f(x)}{x}=\frac{f(y)}{y}=c$ , a constant $\Rightarrow f(x)=c x \Rightarrow f(x)=-x$ from the given condition. $y=f(x)=-x$ does not intersect the parabola $y=x^{2}+1$ because $x^{2}-x+1=0$ has no real solutions.)

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