The Extremal Principle
Look at the largest, the smallest, the closest, the first.
Take the extreme object — the minimal counterexample, the largest number on the board, the closest pair of points — and squeeze it. The extreme object has special properties precisely because nothing is beyond it, and those properties frequently yield a contradiction or force the structure you need.
Infinite descent is the arithmetic version: if every solution generates a strictly smaller one, no solution exists. Vieta jumping (IMO 1988/6) is exactly a descent on the minimal solution.
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