TR-MO 2022

In triangle $ABC$ , $M$ is the midpoint of side $BC$ , the bisector of angle $BAC$ intersects $BC$ and $(ABC)$ at $K$ and $L$ , respectively. If the circle with diameter $[BC]$ is tangent to the external angle bisector of angle $BAC$ , prove that this circle is tangent to $(KLM)$ as well.

For positive integers $k$ and $n$ , we know $k \geq n!$ . Prove that $\phi (k) \geq (n-1)!$

Let $a_1, a_2, \cdots, a_{2022}$ be nonnegative real numbers such that $a_1+a_2+\cdots +a_{2022}=1$ . Find the maximum number of ordered pairs $(i, j)$ , $1\leq i,j\leq 2022$ , satisfying $$a_i^2+a_j\ge \frac 1{2021}.$$

For which real numbers $a$ , there exist pairwise different real numbers $x, y, z$ satisfying $$\frac{x^3+a}{y+z}=\frac{y^3+a}{x+z}=\frac{z^3+a}{x+y}= -3.$$

In triangle $ABC$ , $90^{o}> \angle A> \angle B> \angle C$ . Let the circumcenter and orthocenter of the triangle be $O$ and $H$ . $OH$ intersects $BC$ at $T$ and the circumcenter of $(AHO)$ is $X$ . Prove that the reflection of $H$ over $XT$ lies on the circumcircle of triangle $ABC$ .

In a school with $2022$ students, either a museum trip or a nature trip is organized every day during a holiday. No student participates in the same type of trip twice, and the number of students attending each trip is different. If there are no two students participating in the same two trips together, find the maximum number of trips held.