11th Putnam 1951

Problem B4

Can we find four distinct concentric circles all touching an ellipse?



Answer: yes, we can find four such circles for any ellipse except the circle.

We show first that we can find four such circles for any non-circular ellipse. Let the ellipse have centre O, major axis AB (length 2a) and minor axis CD (length 2b). Take a point P with OP < (a - b)/2. Then using the triangle inequality we have PA > OA - OP = (a + b)/2, PB > OB - OP = (a + b)/2. PC < OC + OP = (a + b)/2, PD < OD + OP = (a + b)/2. Put k = (a + b)/2. Then, if Q is a point on the ellipse, (PQ - k) is positive at A and B and negative at C and D. Hence there must be at least four points Q on the ellipse at which it is stationary and hence at which PQ is normal to the ellipse. There is a point A' near A which is a maximum and hence has PA' > k. Similarly, there is a point B' near B with PB' > k. There is a point C' near C with PC' < k and a point D' near D with PD' < k. Let us assume that P is in the quadrant AOC and not on either axis. The reflection D'' of D' in the line AB lies at or near C' and hence has PD'' >= PC' (which is known to be a minimum). Put P lies closer to D'' than to D'. Hence PC' < PD'. A similar argument shows that PB' > PA'. So we have PB' > PA' > k > PD' > PC'. Thus the circles centre P through A', B', C' and D' are all distinct and all touch the ellipse.

Note that we cannot find 4 circles in the case when the ellipse is circular. For given a point P and a circle E centre O, a circle centre P can only touch E on the line OP, but that implies there can only be two such circles.



11th Putnam 1951

© John Scholes
5 Mar 2002