CGMO 2019

Let $ABCD$ be a cyclic quadrilateral with circumcircle $\odot O.$ The lines tangent to $\odot O$ at $A,B$ intersect at $L.$ $M$ is the midpoint of the segment $AB.$ The line passing through $D$ and parallel to $CM$ intersects $\odot (CDL)$ at $F.$ Line $CF$ intersects $DM$ at $K,$ and intersects $\odot O$ at $E$ (different from point $C$ ). Prove that $EK=DK.$

For a sequence, one can perform the following operation: select three adjacent terms $a,b,c,$ and change it into $b,c,a.$ Determine all the possible positive integers $n\geq 3,$ such that after finite number of operation, the sequence $1,2,\cdots, n$ can be changed into $n,n-1,\cdots,1$ finally.

Let $p$ be a prime number such that $p\mid (2^{2019}-1) .$ The sequence $a_1,a_2,...,a_n$ satisfies the following conditions: $a_0=2, a_1=1 ,a_{n+1}=a_n+\frac{p^2-1}{4}a_{n-1}$ $(n\geq 1).$ Prove that $p\nmid (a_n+1),$ for any $n\geq 0.$

Let $0\leq x_1\leq x_2\leq \cdots \leq x_n\leq 1$ $(n\geq 2).$ Prove that $$\sqrt[n]{x_1x_2 \cdots x_n}+ \sqrt[n]{(1-x_1)(1-x_2)\cdots (1-x_n)}\leq \sqrt[n]{1-(x_1- x_n)^2}.$$

Let $DFGE$ be a cyclic quadrilateral. Line $DF$ intersects $EG$ at $C,$ and line $FE$ intersects $DG$ at $H.$ $J$ is the midpoint of $FG.$ The line $\ell$ is the reflection of the line $DE$ in $CH,$ and it intersects line $GF$ at $I.$ Prove that $C,J,H,I$ are concyclic.

For a tournament with $8$ vertices, if from any vertex it is impossible to follow a route to return to itself, we call the graph a good graph. Otherwise, we call it a bad graph. Prove that $(1)$ there exists a tournament with $8$ vertices such that after changing the orientation of any at most $7$ edges of the tournament, the graph is always abad graph; $(2)$ for any tournament with $8$ vertices, one can change the orientation of at most $8$ edges of the tournament to get a good graph. (A tournament is a complete graph with directed edges.)