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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.

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