ÖMO 2023

Let $a, b, c, d$ be positive reals strictly smaller than $1$ , such that $a+b+c+d=2$ . Prove that $$\sqrt{(1-a)(1-b)(1-c)(1-d)} \leq \frac{ac+bd}{2}.$$

Given is a triangle $ABC$ . The points $P, Q$ lie on the extensions of $BC$ beyond $B, C$ , respectively, such that $BP=BA$ and $CQ=CA$ . Prove that the circumcenter of triangle $APQ$ lies on the angle bisector of $\angle BAC$ .

Given a positive integer $n$ , find the proportion of the subsets of $\{1,2, \ldots, 2n\}$ such that their smallest element is odd.

Find all pairs of positive integers $(n, k)$ satisfying the equation $$n!+n=n^k.$$