Problem 5

Let

P_1,P_2,\ldots,P_{1993}=P_0

be distinct points in the

xy

-plane with integer coordinates. Assume that no point other than

P_i

and

P_{i+1}

on the segment

P_iP_{i+1}

has both coordinates integers, for

i=0,1,\ldots,1992

. Prove that for some

i

0\le i\le1992

, there is a point

Q=(q_x,q_y)

P_iP_{i+1}

such that both

2q_x

and

2q_y

are odd integers.

Step 2 of 3: Find adjacent points of the same class

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P_i=(a,b),\ P_{i+1}=(c,d)\text{ have the same parity for some }i

Detailed analysis

Among the odd number of points $P_0,P_1,\ldots,P_{1992}$ , two consecutive points must have the same one of the two parities; otherwise the classes would alternate and the final class would differ from the first.