# 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 $$\mathbb{Z}$$:

• $$a^3+2b^3 = 4c^3$$

• $$2a^2+3b^2 = c^2+6d^2$$

• $$x^2 + y^2 + z^2 = 2xyz$$

• $$x^4+y^4 = z^2$$

• I wonder why should this be closed? – Maria Mazur Apr 6 at 20:30
• FWIW I voted to leave this open. However, I do have the misgiving that this type of a question does not have a "correct" answer. I guess opinions differ whether that makes a question unsuitable. It sounds like you welcome many answers, so may be using the "meta"tag big-list would be a good idea? – Jyrki Lahtonen Apr 6 at 21:03

## 1 Answer

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...