GR-HMS 2020

Find all non constant polynomials $P(x),Q(x)$ with real coefficients such that: $P((Q(x))^3)=xP(x)(Q(x))^3$

Given a line segment $AB$ and a point $C$ lies inside it such that $AB=3 \cdot AC$ . Construct a parallelogram $ACDE$ such that $AC=DE=CE>AR$ . Let $Z$ be a point on $AC$ such that $\angle AEZ=\angle ACE =\omega$ . Prove that the line passing through point $B$ and perpendicular on side $EC$ , and the line passing through point $D$ and perpendicular on side $AB$ , intersect on point , let it be $K$ , lying on line $EZ$ .

On the board there are written in a row, the integers from $1$ until $2030$ (included that) in an increasing order. We have the right of ''movement'' $K$ : We choose any two numbers $a,b$ that are written in consecutive positions and we replace the pair $(a,b)$ by the number $(a-b)^{2020}$ . We repeat the movement $K$ , many times until only one number remains written on the board. Determine whether it would be possible, that number to be: (i) $2020^{2020}$ (ii) $2021^{2020}$

Find all values of the positive integer $k$ that has the property: There are no positive integers $a,b$ such that the expression $A(k,a,b)=\frac{a+b}{a^2+k^2b^2-k^2ab}$ is a composite positive number.