Let $ABC$ be a right triangle with $m(\widehat {C}) = 90^\circ$, and $D$ be its incenter. Let $N$ be the intersection of the line $AD$ and the side $CB$. If $|CA|+|AD|=|CB|$, and $|CN|=2$, then what is $|NB|$?

Find all solutions of the equation $4^x+3^y=z^2$ in positive integers.

There are $24$ cups on a table. In the beginning, only three of them placed upside-down. At each step, we are turning four cups. Can we turn all the cups right-side up in at most $100$ steps?