51st Putnam 1990

Problem B6

C is a non-empty, closed, bounded, convex subset of the plane. Given a support line L of C and a real number 0 ≤ α ≤ 1, let Bα be the band parallel to L, situated midway between L and the parallel support line on the other side of C, and of width α times the distance between the two support lines. What is the smallest α such that ∩ Bα contains a point of C, where the intersection is taken over all possible directions for the support line L?



Answer: 1/3

The key is to consider the centroid of C. First, we need the result that the centroid cannot lie too far from the midline between two parallel support lines. To be more precise, it can lie at most 1/3 of the way from the midline to the support lines. Let L and L' be parallel support lines, with P a point in the intersection of L and C. Take a triangle PQR with base QR on L' so that the area above PQ in C equals the area below PQ not in C and similarly for PR. Then the centroid of C is above or level with that of PQR and so cannot be below 1/3 of the way from the midline towards L'. Similarly, it cannot be above 1/3 of the way from the midline towards L.

For a triangle the intersection is empty if α < 1/3. If α ≥1/3, then the intersection contains at least the centroid.



51st Putnam 1990

© John Scholes
1 Jan 2001