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. [asy][asy] unitsize(1 cm); int i, j; for (i = 0; i <= 3; ++i) { draw((i,0)--(i,3)); draw((0,i)--(3,i)); } draw((1,1)--(2,2)--(1,3)--(0,2)--cycle); for (i = 0; i <= 3; ++i) { for (j = 0; j <= 3; ++j) { dot((i,j)); }} [/asy][/asy] Calculate the number of different squares that can be formed in a lattice of $100\times 100$ points.

Given the isosceles triangle $ABC$ with $| AB | = | AC | = 10$ and $| BC | = 15$. Let points $P$ in $BC$ and $Q$ in $AC$ chosen such that $| AQ | = | QP | = | P C |$. Calculate the ratio of areas of the triangles $(PQA): (ABC)$.

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