D is a disk. Show that we cannot find congruent sets A, B with A ∩ B = ∅, A ∪ B = D.
It is easy to find two congruent sets which overlap only at the centre O of the disk, so this suggests that the centre is the key.
Wlog we may assume O is in A and that the radius is 1. Let O' be the corresponding point in B. Let PQ be the diameter perpendicular to OO'. Then PO' and QO' are both greater than 1. But OX ≤ 1 for any point X in A, so O'Y ≤ 1 for any point Y in B. So P and Q must both be in A. Let P' and Q' be the corresponding points in B. Then P'Q' = PQ = 2, so P'Q' must be a diameter. But O'P' = OP = 1 and O'Q' = OQ = 1, so O' must be the midpoint of P'Q' and hence the centre of the disk. So O' = O. Contradiction.
25th Putnam 1964
© John Scholes
5 Feb 2002