Problem
ALG-B1-M09-P023 Squares around a center
#23
★★★★★ Level 5 of 5
Find all integer pairs \((x,y)\) such that \(x^2+y^2=3(x+y)\).
Multiply by \(4\) and complete squares \(2x-3\), \(2y-3\).
Multiply by \(4\): \(4x^2-12x+4y^2-12y=0\). Then \((2x-3)^2+(2y-3)^2=18\).
The only odd squares summing to \(18\) are \(9\) and \(9\). Hence \(2x-3=\pm3\), \(2y-3=\pm3\). Thus \(x,y\in\{0,3\}\). All solutions are \((0,0),(0,3),(3,0),(3,3)\).
A strong problem on completing squares and integer residues.