CGMO 2022

Consider all the real sequences $x_0,x_1,\cdots,x_{100}$ satisfying the following two requirements: (1) $x_0=0$ ; (2)For any integer $i,1\leq i\leq 100$ ,we have $1\leq x_i-x_{i-1}\leq 2$ . Find the greatest positive integer $k\leq 100$ ,so that for any sequence $x_0,x_1,\cdots,x_{100}$ like this,we have $$x_k+x_{k+1}+\cdots+x_{100}\geq x_0+x_1+\cdots+x_{k-1}.$$

Let $n$ be a positive integer. There are $3n$ women's volleyball teams in the tournament, with no more than one match between every two teams (there are no ties in volleyball). We know that there are $3n^2$ games played in this tournament.

In triangle $ABC,AB>AC,I$ is the incenter, $AM$ is the midline. The line crosses $I$ and is perpendicular to $BC$ intersect $AM$ at point $L$ , and the symmetry of $I$ with respect to point $A$ is $J$ Prove that: $\angle ABJ= \angle LBI$ .

Given a prime number $p\ge 5$ . Find the number of distinct remainders modulus $p$ of the product of three consecutive positive integers.

Two points $K$ and $L$ are chosen inside triangle $ABC$ and a point $D$ is chosen on the side $AB$ . Suppose that $B$ , $K$ , $L$ , $C$ are concyclic, $\angle AKD = \angle BCK$ and $\angle ALD = \angle BCL$ . Prove that $AK = AL$ .

Find all integers $n$ satisfying the following property. There exist nonempty finite integer sets $A$ and $B$ such that for any integer $m$ , exactly one of these three statements below is true: (a) There is $a \in A$ such that $m \equiv a \pmod n$ , (b) There is $b \in B$ such that $m \equiv b \pmod n$ , and (c) There are $a \in A$ and $b \in B$ such that $m \equiv a + b \pmod n$ .

Let $n \geqslant 3$ be integer. Given convex $n-$ polygon $\mathcal{P}$ . A $3-$ coloring of the vertices of $\mathcal{P}$ is called nice such that every interior point of $\mathcal{P}$ is inside or on the bound of a triangle formed by polygon vertices with pairwise distinct colors. Determine the number of different nice colorings. (Two colorings are different as long as they differ at some vertices.)

Let $x_1, x_2, \ldots, x_{11}$ be nonnegative reals such that their sum is $1$ . For $i = 1,2, \ldots, 11$ , let $$y_i = \begin{cases} x_{i} + x_{i + 1}, & i \textup{odd} , x_{i} + x_{i + 1} + x_{i + 2}, & i \textup{even} ,\end{cases}$$ where $x_{12} = x_{1}$ . And let $F (x_1, x_2, \ldots, x_{11}) = y_1 y_2 \ldots y_{11}$ . Prove that $x_6 < x_8$ when $F$ achieves its maximum.