Let $P_0(x)=x^3-4x$. Sequence of polynomials is defined as following: $P_{n+1}=P_n(1+x)P_n(1-x)-1$. Prove that $x^{2016}|P_{2016}(x)$.

Let $ABCD$ be a square with side $4$. Find, with proof, the biggest $k$ such that no matter how we place $k$ points into $ABCD$, such that they are on the interior but not on the sides, we always have a square with sidr length $1$, which is inside the square $ABCD$, such that it contains no points in its interior(they can be on the sides).

Let $w(x)$ be largest odd divisor of $x$. Let $a,b$ be natural numbers such that $(a,b)=1$ and $a+w(b+1)$ and $b+w(a+1)$ are powers of two. Prove that $a+1$ and $b+1$ are powers of two.