Show that if four distinct points of the curve y = 2x4 + 7x3 + 3x - 5 are collinear, then their average x-coordinate is some constant k. Find k.
Answer: - 7/8
Suppose the common line is y = ax + b, the the x-coordinates satisfy 2x4 + 7x3 + (3 - a)x - (5 + b) = 0. This has at most 4 distinct roots. The arithmetic mean of the roots is 1/4 (-7/2) = -7/8.
The only other possibility is that the line is x = a, but that only meets the curve in one point.
38th Putnam 1977
© John Scholes
30 Nov 1999