SCHWEITZER 1983 Problem 3

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable, $2 \pi$ -periodic even function. Prove that if $$f''(x)+f(x)=\frac{1}{f(x+ 3 \pi /2 )}$$ holds for every $x$ , then $f$ is $\pi /2$ -periodic.

Z. Szabo, J. Terjeki