FKMO 2020

Let $ABCD$ be an isosceles trapezoid such that $AB \parallel CD$ and $\overline{AD}=\overline{BC}, \overline{AB}>\overline{CD}$ . Let $E$ be a point such that $\overline{EC}=\overline{AC}$ and $EC \perp BC$ , and $\angle ACE<90^{\circ}$ . Let $\Gamma$ be a circle with center $D$ and radius $DA$ , and $\Omega$ be the circumcircle of triangle $AEB$ . Suppose that $\Gamma$ meets $\Omega$ again at $F(\neq A)$ , and let $G$ be a point on $\Gamma$ such that $\overline{BF}=\overline{BG}$ . Prove that the lines $EG, BD$ meet on $\Omega$ .

There are $2020$ groups, each of which consists of a boy and a girl, such that each student is contained in exactly one group. Suppose that the students shook hands so that the following conditions are satisfied:

• boys didn't shake hands with boys, and girls didn't shake hands with girls;
• in each group, the boy and girl had shake hands exactly once;
• any boy and girl, each in different groups, didn't shake hands more than once;
• for every four students in two different groups, there are at least three handshakes.

Prove that one can pick $4038$ students and arrange them at a circular table so that every two adjacent students had shake hands.

Find all $f: \mathbb{Q}_{+} \rightarrow \mathbb{R}$ such that $$f(x)+f(y)+f(z)=1$$ holds for every positive rationals $x, y, z$ satisfying $x+y+z+1=4xyz$ .

Do there exist two positive reals $\alpha, \beta$ such that each positive integer appears exactly once in the following sequence? $$2020, [\alpha], [\beta], 4040, [2\alpha], [2\beta], 6060, [3\alpha], [3\beta], \cdots$$ If so, determine all such pairs; if not, prove that it is impossible.

Let $ABC$ be an acute triangle such that $\overline{AB}=\overline{AC}$ . Let $M, L, N$ be the midpoints of segment $BC, AM, AC$ , respectively. The circumcircle of triangle $AMC$ , denoted by $\Omega$ , meets segment $AB$ at $P(\neq A)$ , and the segment $BL$ at $Q$ . Let $O$ be the circumcenter of triangle $BQC$ . Suppose that the lines $AC$ and $PQ$ meet at $X$ , $OB$ and $LN$ meet at $Y$ , and $BQ$ and $CO$ meets at $Z$ . Prove that the points $X, Y, Z$ are collinear.

Find all positive integers $n$ such that $6(n^4-1)$ is a square of an integer.