KMO 2020

Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$x^2f(x)+yf(y^2)=f(x+y)f(x^2-xy+y^2)$$ for all $x,y\in\mathbb{R}$ .

$H$ is the orthocenter of an acute triangle $ABC$ , and let $M$ be the midpoint of $BC$ . Suppose $(AH)$ meets $AB$ and $AC$ at $D,E$ respectively. $AH$ meets $DE$ at $P$ , and the line through $H$ perpendicular to $AH$ meets $DM$ at $Q$ . Prove that $P,Q,B$ are collinear.

There are n boys and m girls at Daehan Mathematical High School. Let $d(B)$ a number of girls who know Boy $B$ each other, and let $d(G)$ a number of boys who know Girl $G$ each other. Each girl knows at least one boy each other. Prove that there exist Boy $B$ and Girl $G$ who knows each other in condition that $\frac{d(B)}{d(G)}\ge\frac{m}{n}$ .

Find a pair of coprime positive integers $(m,n)$ other than $(41,12)$ such that $m^2-5n^2$ and $m^2+5n^2$ are both perfect squares.

For some positive integer $n$ , there exists $n$ different positive integers $a_1, a_2, ..., a_n$ such that $(1)$ $a_1=1, a_n=2000$ $(2)$ $\forall i\in \mathbb{Z}$ $s.t.$ $2\le i\le n, a_i -a_{i-1}\in \{-3,5\}$ Determine the maximum value of n.

Let $ABCDE$ be a convex pentagon such that quadrilateral $ABDE$ is a parallelogram and quadrilateral $BCDE$ is inscribed in a circle. The circle with center $C$ and radius $CD$ intersects the line $BD, DE$ at points $F, G(\neq D)$ , and points $A, F, G$ is on line l. Let $H$ be the intersection point of line $l$ and segment $BC$ . Consider the set of circle $\Omega$ satisfying the following condition.

Circle $\Omega$ passes through $A, H$ and intersects the sides $AB, AE$ at point other than $A$ .

Let $P, Q(\neq A)$ be the intersection point of circle $\Omega$ and sides $AB, AE$ . Prove that $AP+AQ$ is constant.