Hello 2016

Geometry Level 3

For real numbers $a,b,c,d$ and $e$ , we are given that $\sin^{-1}a + \sin^{-1}b + \sin^{-1}c + \sin^{-1}d + \sin^{-1}e = \dfrac{5\pi}2$ .

For real $x$ , let $f$ denote the maximum value of $3\sin x + 4\cos x$ and $g$ denote the minimum positive value of $\tan x + \dfrac1{\tan x}$ .

Compute $\dfrac{a^{2016} + b^{2016} + c^{2016} + d^{2016} + e^{2016} + 5g}{abcdef}$ .

The answer is 3.

1 solution

Aniket Sanghi
Feb 3, 2016

for inverse each must have maximum value $\frac { \pi }{ 2 }$ ....hence $a=b=c=d=e=1$ also for max value of $3 sin x + 4 cos x = \sqrt { 3^2 +4^2 }=5$ ........then min value of $tan x + \frac { 1 }{ tan x }=2$ by AM>=GM...HENCE PUT THE VALUES AND GET THE ANSWER

Yup the problem was not as difficult as it was appearing... BTW Nice problem..

Rishabh Jain - 5 years, 4 months ago

Thank you ...It is a mixture of three major concepts

Aniket Sanghi - 5 years, 4 months ago

Typo its the AM-GM Inequality not AP GP :)

Prakhar Bindal - 4 years, 6 months ago

Hello 2016

The answer is 3.

1 solution

0 pending reports