Putnam 1996

Problem A2

Two circles have radii 1 and 3 and centers a distance 10 apart. Find the locus of all points which are the midpoint of a segment with one end on each circle.



Answer: Let O1, O2 be the centers of C1, C2 and O the midpoint of O1O2. The locus is the annulus radii 1 and 2, center O.


Fix Y on the C2. Then the locus of the midpoint is a circle radius 1/2, center N, the midpoint of XO1. Now vary X. The locus of N is a circle radius 1 1/2 center O. Hence the locus of M is the area swept out by the circle radius 1/2 as its center moves around the circle radius 1 1/2. This is an annulus radii 1 and 2.



Putnam 1996

© John Scholes
12 Dec 1998