Combinatorics problems that can be solved via infinite descent I'm looking for high school problems that can be solved with the method of infinite descent. Usually, those problems are from number theory, but I would be very happy if someone could provide a problem(s) from combinatorics and/or any other field of mathematics. Here are some problems from number theory:

Prove that a following equations have no nontrivial solutions in $\mathbb{Z}$:

*

*$a^3+2b^3 = 4c^3$


*$2a^2+3b^2 = c^2+6d^2$


*$x^2 + y^2 + z^2 = 2xyz$


*$x^4+y^4 = z^2$


What about this one :
Let $(a,b,c)$ in $\mathbb{N}$ such that $(a^2+b^2)/(1+ab) =c $
Prove that $c = p^2$ with $p \in \mathbb{N}$
I don't have a proof though...
