6th Putnam 1946

Problem B1

Two circles C1 and C2 intersect at A and B. C1 has radius 1. L denotes the arc AB of C2 which lies inside C1. L divides C1 into two parts of equal area. Show L has length > 2.



Let O1 be the center of C1, and O2 the center of C2. Let the line O1O2 meet the arc AB of C2 at P. If P lies between O1 and O2, then the tangent to C2 at P divides C1 into two unequal parts and the area C1 ∩ C2 lies inside the smaller part. Contradiction. So O1 must lie between P and O2. But now the arc AP is greater than the segment AP, which is greater than AO1 = 1. Hence L > 2.



6th Putnam 1946

© John Scholes
14 Dec 1999