Question: Find all integers x, y, z such that > x^2 + y^2 = -z^2. > Why it’s impossible: For any integer, a square is always nonnegative, so: * x^2 \ge 0 * y^2 \ge 0 * z^2 \ge 0 The left side, x^2 + y^2, is therefore at least 0, while the right side, -z^2, is at most 0. They can only be equal if both sides are 0, which happens only when: x = y = z = 0. So if the problem instead asks for nonzero integers, it has no solution. If you want something that looks solvable but is actually impossible (an olympiad-style trick question), try this: Challenge: Find positive integers a, b, c satisfying > a^2 + b^2 + c^2 = -1. > This is impossible because the left side is always nonnegative, so it can never equal -1.