CGMO 2021

Let $n \in \mathbb{N}^+,$ $x_1,x_2,...,x_{n+1},p,q\in \mathbb{R}^+$ , $p<q$ and $x^p_{n+1}>\sum_{i=1}^{n}x^p_{i}.$ Prove that $(1)x^q_{n+1}>\sum_{i=1}^{n}x^q_{i};$ $(2)\left(x^p_{n+1}-\sum_{i=1}^{n}x^p_{i}\right)^{\frac{1}{p}}<\left(x^q_{n+1}-\sum_{i=1}^{n}x^q_{i}\right)^{\frac{1}{q}}.$

In acute triangle $ABC$ ( $AB \neq AC$ ), $I$ is its incenter and $J$ is the $A$ -excenter. $X, Y$ are on minor arcs $\widehat{AB}$ and $\widehat{AC}$ respectively such that $\angle{AXI}=\angle{AYJ}=90^{\circ}$ . $K$ is on line $BC$ such that $KI=KJ$ . Proof that line $AK$ bisects $\overline{XY}$ .

Find the smallest positive integer $n$ , such that one can color every cell of a $n \times n$ grid in red, yellow or blue with all the following conditions satisfied: (1) the number of cells colored in each color is the same; (2) if a row contains a red cell, that row must contain a blue cell and cannot contain a yellow cell; (3) if a column contains a blue cell, it must contain a red cell but cannot contain a yellow cell.

Call a sequence of positive integers $(a_n)_{n \ge 1}$ a "CGMO sequence" if $(a_n)_{n \ge 1}$ strictly increases, and for all integers $n \ge 2022$ , $a_n$ is the smallest integer such that there exists a non-empty subset of $\{a_{1}, a_{2}, \cdots, a_{n-1} \}$ $A_n$ where $a_n \cdot \prod\limits_{a \in A_n} a$ is a perfect square.

Proof that if $4$ numbers (not necessarily distinct) are picked from $\{1, 2, \cdots, 20\}$ , one can pick $3$ numbers among them and can label these $3$ as $a, b, c$ such that $ax \equiv b \;(\bmod c)$ has integral solutions.

Given a finite set $S$ , $P(S)$ denotes the set of all the subsets of $S$ . For any $f:P(S)\rightarrow \mathbb{R}$ ,prove the following inequality: $$\sum_{A\in P(S)}\sum_{B\in P(S)}f(A)f(B)2^{\left| A\cap B \right|}\geq 0.$$

In an acute triangle $ABC$ , $AB \neq AC$ , $O$ is its circumcenter. $K$ is the reflection of $B$ over $AC$ and $L$ is the reflection of $C$ over $AB$ . $X$ is a point within $ABC$ such that $AX \perp BC, XK=XL$ . Points $Y, Z$ are on $\overline{BK}, \overline{CL}$ respectively, satisfying $XY \perp CK, XZ \perp BL$ .

Proof that $B, C, Y, O, Z$ lie on a circle.

Let $m, n$ be positive integers, define: $f(x)=(x-1)(x^2-1)\cdots(x^m-1)$ , $g(x)=(x^{n+1}-1)(x^{n+2}-1)\cdots(x^{n+m}-1)$ .

Show that there exists a polynomial $h(x)$ of degree $mn$ such that $f(x)h(x)=g(x)$ , and its $mn+1$ coefficients are all positive integers.