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Yes, it's quite simple, though I didn't know how to solve it when I took the test (the 'simple' solutions tend to elude me). The idea of completing the square was somewhat novel to me, back then (i.e., May of this year). ;-_-

metafor wrote: If the sides of a triangle have lengths $ a, b, c$ such that $ a + b - c = 2$ and $ 2ab - c^{2} = 4$, then prove that the triangle is equilateral. Proof. Otherwise, $ 2ab - 4 = (a + b - 2)^2$ $ \implies$ $ a^2 + b^2 - 4a - 4b + 8 = 0$ $ \implies$ $ (a - 2)^2 + (b - 2)^2 = 0$ $ \implies$ $ a = b = 2 = c$ .