MathDB

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.

Problem Statement