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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 $\mathbb{Z}$:

  • $a^3+2b^3 = 4c^3$

  • $2a^2+3b^2 = c^2+6d^2$

  • $x^2 + y^2 + z^2 = 2xyz$

  • $x^4+y^4 = z^2$

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    $\begingroup$ I wonder why should this be closed? $\endgroup$ – Maria Mazur Apr 6 at 20:30
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    $\begingroup$ 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? $\endgroup$ – Jyrki Lahtonen Apr 6 at 21:03
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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...

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