Math Gold Medalist

Lor

2023 USAJMO

Problem 1

Find all triples of positive integers $(x,y,z)$ that satisfy the equation\begin{align*} 2(x+y+z+2xyz)^2=(2xy+2yz+2zx+1)^2+2023 \end{align*}


Identities

Change of Variables

Factorization

Expand both sides

Changing variable: a=2x^2, b=2y^2, c=2z^2

(a-1)(b-1)(c-1)=2023

Prime factorize 2023

Factorize a^3+b^3+c^3-3abc

Factorize (a^2)b+(b^2)a+(a^2)c+(c^2)a+(b^2)c+(c^2)b+2abc

   Solution