JBMO 2014

Find all distinct prime numbers $p, q$ and $r$ such that

$$3 p^{4}-5 q^{4}-4 r^{2}=26$$

Consider an acute triangle $A B C$ with area S. Let $C D \perp A B (D \in A B)$, $D M \perp A C (M \in A C)$ and $D N \perp B C (N \in B C)$. Denote by $H_{1}$ and $H_{2}$ the orthocentres of the triangles $M N C$ and $M N D$ respectively. Find the area of the quadrilateral $\mathrm{AH}_{1} \mathrm{BH}_{2}$ in terms of $S$.

Let $a, b, c$ be positive real numbers such that $a b c=1$. Prove that:

$$\left(a+\frac{1}{b}\right)^{2}+\left(b+\frac{1}{c}\right)^{2}+\left(c+\frac{1}{a}\right)^{2} \geq 3(a+b+c+1)$$

When does equality hold?

For a positive integer $n$, two players A and B play the following game: Given a pile of $s$ stones, the players take turn alternatively with A going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a multiple of $n$ stones. The winner is the one who takes the last stone. Assuming both $\mathrm{A}$ and $\mathrm{B}$ play perfectly, for how many values of $s$ the player A cannot win?