## N1

Each letter of the word OHRID corresponds to a different digit belonging to the set $(1,2,3,4,5)$. Decipher the equality $(O+H+R+I+D)^{2}:(O-H-R+I+D)=O^{H^{R^{I_{D}^{D}}}}$.


Solution. Since $O, H, R, I$ and $D$ are distinct numbers from $\{1,2,3,4,5\}$, we have $O+H+R+I+D=15$ and $O-H-R+I+D=O+H+R+I+D-2(H+R)15$ and divides 225 , which is only possible for $O^{H^{R^{I^{D}}}}=25$ (must be a power of three or five). This implies that $O=5, H=2$ and $R=1$. It's easy to check that both $I=3, D=4$ and $I=4, D=3$ satisfy the stated equation.

## $\mathbf{N} 2$

Find all triples $(p, q, r)$ of distinct primes $p, q$ and $r$ such that

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