IMO 1967 LL ITA28

Origin: ITA

Problem

Find values of the parameter u for which the expression y = tan(x −u) + tan x + tan(x + u) tan(x −u) tan x tan(x + u) does not depend on x.

Solution

The given expression can be transformed into y = 4 cos 2u + 2 cos 2u −cos 2x −3. It does not depend on x if and only if cos 2u = −1/2, i.e., u = \pm\pi/3 + k\pi for some k \inZ.