FKMO 2022

There are $n$ boxes $A_1, ..., A_n$ with non-negative number of pebbles inside it(so it can be empty). Let $a_n$ be the number of pebbles in the box $A_n$ . There are total $3n$ pebbles in the boxes. From now on, Alice plays the following operation.

In each operation, Alice choose one of these boxes which is non-empty. Then she divide this pebbles into $n$ group such that difference of number of pebbles in any two group is at most 1, and put these $n$ group of pebbles into $n$ boxes one by one. This continues until only one box has all the pebbles, and the rest of them are empty. And when it's over, define $Length$ as the total number of operations done by Alice.

Let $f(a_1, ..., a_n)$ be the smallest value of $Length$ among all the possible operations on $(a_1, ..., a_n)$ . Find the maximum possible value of $f(a_1, ..., a_n)$ among all the ordered pair $(a_1, ..., a_n)$ , and find all the ordered pair $(a_1, ..., a_n)$ that equality holds.

A function $g \colon \mathbb{R} \to \mathbb{R}$ is given such that its range is a finite set. Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ that satisfies $$2f(x+g(y))=f(2g(x)+y)+f(x+3g(y))$$ for all $x, y \in \mathbb{R}$ .

Let $ABC$ be a scalene triangle with incenter $I$ and let $AI$ meet the circumcircle of triangle $ABC$ again at $M$ . The incircle $\omega$ of triangle $ABC$ is tangent to sides $AB, AC$ at $D, E$ , respectively. Let $O$ be the circumcenter of triangle $BDE$ and let $L$ be the intersection of $\omega$ and the altitude from $A$ to $BC$ so that $A$ and $L$ lie on the same side with respect to $DE$ . Denote by $\Omega$ a circle centered at $O$ and passing through $L$ , and let $AL$ meet $\Omega$ again at $N$ .

Prove that the lines $LD$ and $MB$ meet on the circumcircle of triangle $LNE$ .

Find all positive integers $m$ such that there exists integers $x$ and $y$ that satisfies $$m \mid x^2+11y^2+2022.$$

Set $X$ is called fancy if it satisfies all of the following conditions:

• The number of elements of $X$ is $2022$ .
• Each element of $X$ is a closed interval contained in $[0, 1]$ .
• For any real number $r \in [0, 1]$ , the number of elements of $X$ containing $r$ is less than or equal to $1011$ .

For fancy sets $A, B$ , and intervals $I \in A, J \in B$ , denote by $n(A, B)$ the number of pairs $(I, J)$ such that $I \cap J \neq \emptyset$ . Determine the maximum value of $n(A, B)$ .