Putnam 1994 Problem B2 For which real α does the curve y = x 4 + 9x 3 + α x 2 + 9x + 4 contain four collinear points? Solution
: α  30.375, then y'' has no zeros and hence y' is strictly monotonically increasing. Hence no line cuts the curve in more than two points (if a line gradient k cut the curve at x = a, b and c, with a  b.] It remains to consider the case α = 30.375. In this case y'' has a single zero at x = 2.25. In this case there is no point of inflection. The gradient is monotonic and hence a line can only cut the curve in two points. [We cannot have degeneracy with infinitely many solutions because we are dealing with a polynomial which has at most 4 solutions.] Putnam 1994 © John Scholes jscholes@kalva.demon.co.uk 12 Dec 1998