Problem

Source: Serbia TST 2022, Problem 6 (source: https://dms.rs/wp-content/uploads/2022/05/ZADACI_IZBORNO_IMO_2022.pdf)

Tags: geometry, trapezoid



Let $ABCD$ be a trapezoid with bases $AB,CD$ such that $CD=k \cdot AB$ ($0<k<1$). Point $P$ is such that $\angle PAB=\angle CAD$ and $\angle PBA=\angle DBC$. Prove that $PA+PB \leq \dfrac{1}{\sqrt{1-k^2}} \cdot AB$.