Putnam 1997 Problem A1 ROMN is a rectangle with vertices in that order and RO = 11, OM = 5. The triangle ABC has circumcenter O and its altitudes intersect at R. M is the midpoint of BC, and AN is the altitude from A to BC. What is the length of BC? Solution
: 28. Easy (provided you know some geometry, otherwise slog it out with coordinates). The orthocenter, median and circumcenter are collinear (the Euler line), so X, the intersection of AM and RO is the median. But AX = 2XM, so AR = 2RN = 10. Hence AO 2 = AR 2 + RO 2 = 221. So MC 2 = OC 2 - OM 2 = 221 - 25 = 196, so MC = 14 and BC = 28. Putnam 1997 © John Scholes jscholes@kalva.demon.co.uk 12 Dec 1998