MathDB

Let

A{}

be a point in the Cartesian plane. At each step, Ann tells Bob a number

0\leqslant a\leqslant 1

and he then moves

A{}

in one of the four cardinal directions, at his choice, by a distance of

a{}.

This process cotinues as long as Ann wishes. Amongst every 100 consecutive moves, each of the four possible moves should have been made at least once. Ann's goal is to force Bob to eventually choose a point at a distance greater than 100 from the initial position of

A.{}

Can Ann achieve her goal?
Selected from an Argentine Olympiad

Problem Statement