Find all positive prime numbers $p$, $q$ that satisfy the equation $$p(p^4+p^2+10q)=q(q^2+3)$$

On a $5 \times 5$ board, pieces made up of $4$ squares are placed, as seen in the figure, each covering exactly $4$ squares of the board. The pieces can be rotated or turned over. They can also overlap, but they cannot protrude from the board. Suppose that each square on the board is covered by at most two pieces. Find the maximum number of squares on the board that can be covered (by one or two pieces).

Let $ABC$ be an acute scalene triangle and let $M$ be the midpoint of side $BC$. The angle bisector of the $\angle BAC$, the perpendicular bisector of the side $AB$ and the perpendicular bisector of the side $AC$ define a new triangle. Let $H$ be the point of intersection of the three altitudes of this new triangle. Show that $H$ belongs to line segment $AM$.

Determine all natural numbers $n \geqslant 2$ with the property that there are two permutations $(a_1, a_2,... , a_n) $ and $(b_1, b_2,... , b_n)$ of the numbers $1, 2,..., n$ such that $(a_1 + b_1, a_2 +b_2,..., a_n + b_n)$ are consecutive natural numbers.

Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that $$(x^2-y^2)f(xy)=xf(x^2y)-yf(xy^2)$$for all real numbers $x, y$.

Uri has $99$ empty bags and an unlimited number of balls. The weight of each ball is a number of the form $3^n$ where $n$ is an integer that can vary from ball to ball (negative integer exponents are allowed, such as $3^{-4}=\dfrac{1}{81}$, and the exponent $0$, where $3^0=1$). Uri chose a finite number of balls and distributed them into the bags so that all the bags had the same total weight and there were no balls left over. It is known that Uri chose at most $k$ balls of the same weight. Find the smallest possible value of $k$.