Valid Inequality for a Triangle!

Geometry Level 5

( h a t + h b t + h c t 3 ) 1 / t 3 2 ( a t + b t + c t 3 ) 1 / t \large{\left( \dfrac{h_a^t + h_b^t + h_c^t}{3} \right)^{1/t} \leq \dfrac{\sqrt{3}}{2}\left( \dfrac{a^t + b^t + c^t}{3}\right)^{1/t}}

Let a , b , c a,b,c be the sides of a triangle and h a , h b , h c h_a, h_b, h_c respectively be the corresponding altitudes. If the maximum range of validity of the above inequality for t t be: α < t < α \quad \large{-\alpha < t < \alpha} ,

where t 0 t \neq 0 and α = ln ( A ) ln ( B ) \alpha = \dfrac{\ln(A)}{\ln(B)} , and where A , B R A,B \in \mathbb R , find the value of A B \dfrac{A}{B} .


The answer is 3.

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