### 38th Putnam 1977

**Problem A1**

Show that if four distinct points of the curve y = 2x^{4} + 7x^{3} + 3x - 5 are collinear, then their average x-coordinate is some constant k. Find k.

**Solution**

Answer: - 7/8

*Trivial.*

Suppose the common line is y = ax + b, the the x-coordinates satisfy 2x^{4} + 7x^{3} + (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

jscholes@kalva.demon.co.uk

30 Nov 1999