KURSCHAK 2017

Let $ABC$ be a triangle. Choose points $A'$ , $B'$ and $C'$ independently on side segments $BC$ , $CA$ and $AB$ respectively with a uniform distribution. For a point $Z$ in the plane, let $p(Z)$ denote the probability that $Z$ is contained in the triangle enclosed by lines $AA'$ , $BB'$ and $CC'$ . For which interior point $Z$ in triangle $ABC$ is $p(Z)$ maximised?

Do there exist polynomials $p(x)$ and $q(x)$ with real coefficients such that $p^3(x)-q^2(x)$ is linear but not constant?

An $n$ by $n$ table has an integer in each cell, such that no two cells within a row share the same number. Prove that it is possible to permute the elements within each row to obtain a table that has $n$ distinct numbers in each column.