CL-MO 2020

Determine all positive integers $n$ such that the decimal representation of the number $6^n + 1$ has all its digits the same.

The points of this lattice $4\times 4 = 16$ points can be vertices of squares.

Calculate the number of different squares that can be formed in a lattice of $100\times 100$ points.

Determine all three integers $(x, y, z)$ that are solutions of the system $$x + y -z = 6$$ $$x^3 + y^3 -z^3 = 414$$