JBMO 2018

Find all the pairs $(m, n)$ of integers which satisfy the equation

$$m^{5}-n^{5}=16 m n$$

A set $T$ of $n$ three-digit numbers has the following five properties:

(1) No number contains the digit 0 .

(2) The sum of the digits of each number is 9 .

(3) The units digits of any two numbers are different.

(4) The tens digits of any two numbers are different.

(5) The hundreds digits of any two numbers are different.

Find the largest possible value of $n$.

Let $A B C$ be an acute triangle, $A^{\prime}, B^{\prime}$ and $C^{\prime}$ be the reflections of the vertices $A, B$ and $C$ with respect to $B C, C A$, and $A B$, respectively, and let the circumcircles of triangles $A B B^{\prime}$ and $A C C^{\prime}$ meet again at $A_{1}$. Points $B_{1}$ and $C_{1}$ are defined similarly. Prove that the lines $A A_{1}, B B_{1}$ and $C C_{1}$ have a common point.