FKMO 2021

An acute triangle $\triangle ABC$ and its incenter $I$ , circumcenter $O$ is given. The line that is perpendicular to $AI$ and passes $I$ intersects with $AB$ , $AC$ in $D$ , $E$ . The line that is parallel to $BI$ and passes $D$ and the line that is parallel to $CI$ and passes $E$ intersects in $F$ . Denote the circumcircle of $DEF$ as $\omega$ , and its center as $K$ . $\omega$ and $FI$ intersect in $P$ ( $\neq F$ ). Prove that $O,K,P$ is collinear.

Positive integer $k(\ge 8)$ is given. Prove that if there exists a pair of positive integers $(x,y)$ that satisfies the conditions below, then there exists infinitely many pairs $(x,y)$ . (1) $$ $x\mid y^2-3, y\mid x^2-2$ (2) $$ $gcd\left(3x+\frac{2(y^2-3)}{x},2y+\frac{3(x^2-2)}{y}\right)=k$ $$

Let $P$ be a set of people. For two people $A$ and $B$ , if $A$ knows $B$ , $B$ also knows $A$ . Each person in $P$ knows $2$ or less people in the set. $S$ , a subset of $P$ with $k$ people, is called k-independent set of $P$ if any two people in $S$ don't know each other. $X_1, X_2, …, X_{4041}$ are 2021-independent sets of $P$ (not necessarily distinct). Show that there exists a 2021-independent set of $P$ , $\{v_1, v_2, …, v_{2021}\}$ , which satisfies the following condition:

[center] For some integer $1 \le i_1 < i_2 < \cdots < i_{2021} \leq 4041$ , $v_1 \in X_{i_1}, v_2 \in X_{i_2}, \ldots, v_{2021} \in X_{i_{2021}}$ [/center]

Let $G$ be a finite simple graph with maximum degree at most $2$ . Let $X_1, X_2, \ldots, X_{4041}$ be independent sets of size $2021$ (not necessarily distinct). Prove that there exists another independent set $\{v_1, v_2, \ldots, v_{2021}\}$ of size $2021$ and indices $1 \le t_1 < t_2 < \cdots < t_{2021} \le 4041$ such that $v_i \in X_{t_i}$ for all $i$ .

There are $n$ ( $\ge 2$ ) clubs $A_1,A_2,...A_n$ in Korean Mathematical Society. Prove that there exist $n-1$ sets $B_1,B_2,...B_{n-1}$ that satisfy the condition below. (1) $A_1\cup A_2\cup \cdots A_n=B_1\cup B_2\cup \cdots B_{n-1}$ (2) for any $1\le i<j\le n-1$ , $B_i\cap B_j=\emptyset, -1\le\left\vert B_i \right\vert -\left\vert B_j \right\vert\le 1$ (3) for any $1\le i \le n-1$ , there exist $A_k,A_j$ ( $1\le k\le j\le n$ )such that $B_i\subseteq A_k\cup A_j$

The incenter and $A$ -excenter of $\triangle{ABC}$ is $I$ and $O$ . The foot from $A,I$ to $BC$ is $D$ and $E$ . The intersection of $AD$ and $EO$ is $X$ . The circumcenter of $\triangle{BXC}$ is $P$ . Show that the circumcircle of $\triangle{BPC}$ is tangent to the $A$ -excircle if $X$ is on the incircle of $\triangle{ABC}$ .

Find all functions $f,g: \mathbb{R} \to \mathbb{R}$ such that satisfies $$f(x^2-g(y))=g(x)^2-y$$ for all $x,y \in \mathbb{R}$