Since the thread has been just bumped to the front page, I take it as an opportunity to add a link to Zhi-Wei Sun's homepage, where several problems with monetary rewards are listed. Most notably, Zhi-Wei Sun is offering 2400 \$ for his "24-Conjecture" (that any natural number can be written as the sum of squares of four nonnegative integers $x$, $y$, $z$, and $w$ such that both $x$ and $x+24y$ are squares) and 1350 \$ for his "1-3-5-Conjecture" (that any natural number can be written as the sum of squares of four nonnegative integers $x$, $y$, $z$, and $w$ such that $x+3y+5z$ is also a square).