IMO Shortlist 1979 problem 21
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arhiva2. travnja 2012. Let
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
be the number of integral solutions of the equation
![x^2 - y^2 = z^3 - t^3](/media/m/0/e/7/0e7629dcabc3aab2c923f0b7e404b374.png)
satisfying the condition
![0 \leq x, y, z, t \leq 10^6](/media/m/b/6/7/b67c0fdf298d62d0cf8f2b9fb783ab66.png)
, and let
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
be the number of integral solutions of the equation
![x^2 - y^2 = z^3 - t^3 + 1](/media/m/9/b/4/9b4e9dd4625e831a6e04d12f798eb565.png)
satisfying the condition
![0 \leq x, y, z, t \leq 10^6](/media/m/b/6/7/b67c0fdf298d62d0cf8f2b9fb783ab66.png)
. Prove that
%V0
Let $N$ be the number of integral solutions of the equation
$$x^2 - y^2 = z^3 - t^3$$
satisfying the condition $0 \leq x, y, z, t \leq 10^6$, and let $M$ be the number of integral solutions of the equation
$$x^2 - y^2 = z^3 - t^3 + 1$$
satisfying the condition $0 \leq x, y, z, t \leq 10^6$. Prove that $N >M.$
Izvor: Međunarodna matematička olimpijada, shortlist 1979