2025 penguins (675 each of emperors, kings and gentoos) stand in a line from left to right.
They may move by performing the penguin shuffle in which an emperor takes a king's place, a gentoo takes the position vacated by the emperor and the displaced king takes the place vacated by the gentoo.
Prove that, no matter what the starting order, it is possible to place the emperors on the left, kings in the middle and gentoos on the right after a finite number of shuffles.
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Cayley Mathematical Olympiad (2025)
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