MathDB

Let

P_1

P_2

\ldots

P_{1993} = P_0

be distinct points in the

xy

-plane with the following properties: (i) both coordinates of

P_i

are integers, for

i = 1, 2, \ldots, 1993

; (ii) there is no point other than

P_i

and

P_{i+1}

on the line segment joining

P_i

with

P_{i+1}

whose coordinates are both integers, for

i = 0, 1, \ldots, 1992

. Prove that for some

i

0 \leq i \leq 1992

, there exists a point

Q

with coordinates

(q_x, q_y)

on the line segment joining

P_i

with

P_{i+1}

such that both

2q_x

and

2q_y

are odd integers.

Problem Statement