MathDB

A4

Part of 2021 Balkan MO Shortlist

Problems(1)

BMO Shortlist 2021 A4

Source: BMO Shortlist 2021

5/8/2022
Let f,gf, g be functions from the positive integers to the integers. Vlad the impala is jumping around the integer grid. His initial position is x0=(0,0)x_0 = (0, 0), and for every n≥1n \ge 1, his jump is
xn−xn−1=(±f(n),±g(n))x_n - x_{n - 1} = (\pm f(n), \pm g(n)) or (±g(n),±f(n)),(\pm g(n), \pm f(n)),
with eight possibilities in total. Is it always possible that Vlad can choose his jumps to return to his initial location (0,0)(0, 0) infinitely many times when (a) f,gf, g are polynomials with integer coefficients? (b) f,gf, g are any pair of functions from the positive integers to the integers?
Balkanshortlist2021algebraFunctional inequality