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Problem of the Week

Problem E

Terry’s Triangles

Terry is drawing isosceles triangles with side lengths \(a\), \(b\), and \(c\) such that \[\begin{aligned} a&=y-x\\ b&=x+z\\ c&=y-z\end{aligned}\] where \(x\), \(y\), and \(z\) are positive integers and \(x+y+z<10\).
Find all the possible triples \((a,b,c)\) that satisfy this.