Problem

Source: Russian TST 2015, Day 7 P3

Tags: combinatorics, geometry



Let $0<\alpha<1$ be a fixed number. On a lake shaped like a convex polygon, at some point there is a duck and at another point a water lily grows. If the duck is at point $X{}$, then in one move it can swim towards one of the vertices $Y$ of the polygon a distance equal to a $\alpha\cdot XY$. Find all $\alpha{}$ for which, regardless of the shape of the lake and the initial positions of the duck and the lily, after a sequence of adequate moves, the distance between the duck and the lily will be at most one meter.