IMO 1967 LL SWE48

Origin: SWE

Problem

Determine all positive roots of the equation xx = 1/ \sqrt 2.

Solution

Put f(x) = x ln x. The given equation is equivalent to f(x) = f(1/2), which has the solutions x1 = 1/2 and x2 = 1/4. Since the function f is decreasing on (0, 1/e), and increasing on (1/e, +\infty), this equation has no other solutions.