Problem
ALG-B1-M09-P020 Three differences
#20
★★★★☆ Level 4 of 5
Find all integer triples \((x,y,z)\) such that \(x^2+y^2+z^2=xy+yz+zx+2\).
Use the sum of squared differences.
Multiply by \(2\): \((x-y)^2+(y-z)^2+(z-x)^2=4\). Squares of integers sum to \(4\). The only possible square patterns are \(4,0,0\) or \(1,1,2\), but \(2\) is not a square. Hence two differences are \(0\), and the third is \(\pm2\).
Thus two variables are equal, and the third differs from them by \(2\). All solutions are permutations of \((t,t,t+2)\) and \((t,t,t-2)\), where \(t\in\mathbb Z\).
A connection between symmetric algebra and integer checking.