MathDB

The square

0\le x\le 1

0\le y\le 1

is drawn in the plane

Oxy

. A grasshopper sitting at a point

M

with noninteger coordinates outside this square jumps to a new point which is symmetrical to

M

with respect to the leftmost (from the grasshopper’s point of view) vertex of the square. Prove that no matter how many times the grasshopper jumps, it will never reach the distance more than

10 d

from the center

C

of the square, where

d

is the distance between the initial position

M

and the center

C

.
(A Kanel)

Problem Statement