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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