IMO 2008 Shortlist A4

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 : Z → Z...

imomathematicsolympiadshortlistalgebra

IMO 2008 Shortlist A4

Category: Algebra

Problem

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 : Z → Z satisfies f(−1) = 0, f(0) = 1, f(1) = −1 and f(2n

  • m) = f(2n − t(m)) − f(m) for all integers m,n ≥ 0 with 2n

m. Prove that f(3p) ≥ 0 holds for all integers p ≥ 0.