July 14–July 20, 2009

Problem

Consider the two functions f(x) = x2 + 2bx + 1 and g(x) = 2a(x + b), where the variable x and the constants a and b are real numbers. Each pair of constants (a, b) can be considered as the coordinates of a point in an ab-plane.

Let S be the set of all such points (a, b) for which the graphs of y = f(x) and y = g(x) do NOT intersect. Determine the area of S.

Solution

here

List of solvers

Nguyen Manh Tien (high school); Alex Hanson, Bradley Sherman, Matthew Inouye, Huy Hoang-Nguyen, Steve Alkire (undergrad); Mauricio Duarte (graduate); Lloyd Sakazaki, Lawrence Hon, T.R. Mukundan, Alejandro Perez, Akai Phan, Adam Welly, Lincoln Atkinson, Peiyush Jain, Qiyuan Wei, Steve Wilmarth, David Webster, Dean Menezes (outside).

Steve Alkire wins the prize!