A geometry problem by Aakhyat Singh

For angles in a triangle or $A+B+C=180^\circ$ , then $\tan A + \tan B + \tan C = \tan A \tan B \tan C$ . Since $\tan A + \tan B + \tan C = 6$ $\implies \tan A \tan B \tan C = 6$ and $\cot A \cot B \cot C = \dfrac 1{\tan A \tan B \tan C} = \dfrac 16$ . $\implies k = \boxed{1}$ .