Find all $(m, n)$ positive integer pairs such that both $\frac{3n^2}{m}$ and $\sqrt{n^2+m}$ are integers.

We are numbering the rows and columns of a $29 \text{x} 29$ chess table with numbers $1, 2, ..., 29$ in order (Top row is numbered with $1$ and first columns is numbered with $1$ as well). We choose some of the squares in this chess table and for every selected square, we know that there exist at most one square having a row number greater than or equal to this selected square's row number and a column number greater than or equal to this selected square's column number. How many squares can we choose at most?

Let $x, y, z$ be real numbers such that $$x+y+z=2, \;\;\;\; xy+yz+zx=1$$Find the maximum possible value of $x-y$.

Let $X$ be a point on the segment $[BC]$ of an equilateral triangle $ABC$ and let $Y$ and $Z$ be points on the rays $[BA$ and $[CA$ such that the lines $AX, BZ, CY$ are parallel. If the intersection of $XY$ and $AC$ is $M$ and the intersection of $XZ$ and $AB$ is $N$, prove that $MN$ is tangent to the incenter of $ABC$.