If and are constant quantities and are variable subjects to the relation a + b + c = k

MATHEMATICS · Subjective · AoPS · Mathematics · More Inequalities · More Inequalities

Problem

a, b, c

and

k

are constant quantities and

\alpha, \beta, \gamma

are variable subjects to the relation

a \tan \alpha + b \tan \beta + c \tan \gamma = k,

then find the minimum value of

\tan^2 \alpha + \tan^2 \beta + \tan^2 \gamma