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For an integer m, denote by t(m) the unique number in \{1, 2, 3\} such that m + t(m) is a multiple of 3. A function f: \mathbb{Z}\to\mathbb{Z} satisfies f( - 1) = 0, f(0) = 1, f(1) = - 1 and f\left(2^{n} + m\right) = f\left(2^n - t(m)\right) - f(m) for all integers m, n\ge 0 with 2^n > m. Prove that f(3p)\ge 0 holds for all integers p\ge 0.

Proposed by Gerhard Woeginger, Austria

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