Trigonometry Related Part IV

Geometry Level 5

1 a 2 + 1 + 4 b 2 + 4 + 9 c 2 + 9 \frac{1}{a^2+1}+\frac{4}{b^2+4}+\frac{9}{c^2+9}

Given that a , b a,b and c c are positive real numbers satisfying 3 a b + b c + 2 c a = 6 3ab+bc+2ca=6 .

If the maximum value of the expression above can be expressed as A B \frac{A}{B} for coprime positive integers A A and B B , determine A + B A+B .


The answer is 13.

View wiki
Leah Smith by Leah Smith , 25, USA

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

Leah Smith
Jan 1, 2016

The solution for this is short and simple.

15 Helpful 0 Interesting 0 Brilliant 0 Confused

Hi, Happy New year how you thought of such good questions using trigo?

Department 8 - 5 years, 5 months ago

Same method!

archit wagle - 5 years, 5 months ago
Arunava Das
Feb 8, 2018

Substitute a = tan A, b = 2 tan B, c = 3 tan C, and then we will get (tanA tanB+tanB tanC+tanC tanA)=1 which is maximum for A=B=C=30°. So we get the former expression to be [(cos A)^2 + (cos B)^2 +(cos C)^2] by trigonometric manipulation, which reduces to (9/4), which is the answer..

0 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...