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PutnamSuperchallengeIntegrals I: Feynman's Trick

Integrals I: Feynman's Trick · Superchallenge problem

Start with (a,b)=(3,4)(a, b) = (3, 4). A move replaces (a,b)(a, b) by (3a−4b5,4a+3b5)\left(\dfrac{3a - 4b}{5}, \dfrac{4a + 3b}{5}\right). After any number of moves, what is a2+b2a^2 + b^2?