BMO1 2008/09 Question 6 Trigonometry Problem

Question

The obtuse-angled triangle $ABC$ has sides of length $a,b$ and $c$ opposite the angles $\angle A, \angle B$ and $\angle C$ respectively. Prove that $$a^3 \cos A + b^3 \cos B + c^3 \cos C \lt abc.$$

No real idea what to do, so any contributions are appreciated.

Please use this guide for formatting mathematics on this site. It makes things much easier for everyone who reads your question. For instance, I assume that when you write "a3 cos A", you really mean $a^3\cos A$, but for all I know it might be $a \cdot 3\cos A$, or even $a_3 \cos A$, and there is no way for me to tell. — Arthur, Apr 06 '15 at 19:41
let $a$ is max,$x=a^2,y=b^2,z=c^2,x>y+z$, then it become $x^3+y^3+z^3+2xyz>\sum_{cyc} x^2y$ — chenbai, Apr 07 '15 at 04:36

score 1 · Accepted Answer · answered Apr 07 '15 at 06:33

1

it is equivalent to $\frac{\left(a^2-b^2-c^2\right) \left(a^2+b^2-c^2\right) \left(a^2-b^2+c^2\right)}{2 a b c}>0$

answered Apr 07 '15 at 06:33

I'm afraid I don't really see how you can't to this conclusion. – MadChickenMan Apr 07 '15 at 10:29
Sorry, meant to say came* – MadChickenMan Apr 07 '15 at 11:08
Ah I think I may know what you mean, but tell me: how do you make these factorisations. There are so many terms, is it simply a case of having seen the factorisation before? – MadChickenMan Apr 07 '15 at 11:29
i have solved many of such problems in our math-circle in Leipzig with my students two of them participated at the last IMO in Southafrica with a silver medall – Dr. Sonnhard Graubner Apr 07 '15 at 13:08
As someone who is familiar with IMO questions then, may I ask how this question and other BMO questions (if you are familiar with those) compare. – MadChickenMan Apr 07 '15 at 15:39
you can ask me more questions, but i know no all answers – Dr. Sonnhard Graubner Apr 07 '15 at 15:45

1 Answers1