Home » Ask & Discuss » Mathematics. » Algebra « Back to Discussion

Algebra

Forum Expert

Joined:	28 Feb 2007
Post:	2173

14 Mar 2008 14:06:34 IST

0 People liked this

644

Another good one

None

$\text {This is a lovely one} \\ \\ \text {Prove that} \\ \\ \sqrt{a^2-ab+b^2} + \sqrt{b^2-bc+c^2} \geq \sqrt{a^2+ac+c^2}$

Share this article on:

Comments (7)

pardesi .svk

Scorching goIITian

Joined: 7 Dec 2007

Posts: 219

14 Mar 2008 18:13:36 IST

Like 12 people liked this

i am assuming that a,b,c

are positive
consider any point

from there draw rays

and

such that $\angle QPR =\frac{2\pi}{3}$ and QP=a,PR=c

bisect the angle and draw SP=b

.
now the given inequality reduces to the triangle inequality
QS + SR > QR

sandeep ramesh

Blazing goIITian

Joined: 13 Mar 2008

Posts: 1183

14 Mar 2008 18:15:09 IST

Like 0 people liked this

EDITED!

Hari Shankar

Forum Expert

Joined: 28 Feb 2007

Posts: 2173

14 Mar 2008 18:55:44 IST

Like 0 people liked this

You chaps from AOPS are downright ruffians. good job fellas!

sreeraman nagasubramaniyan

Blazing goIITian

Joined: 17 Jan 2008

Posts: 510

14 Mar 2008 19:25:22 IST

Like 0 people liked this

(a²+b²)/2 >= ab

=> ab<=(a²+b²)/2

(a²-ab+b²) >=

(a² -(a²+b²)/2 +b²) =

((a²+b²)/2)

similarly

(b²-bc+c²) >=

((b²+c²)/2)

using AM of mth power >= mth power of AM

(

((a²+b²)/2) +

((b²+c²)/2))/2 > = [((a²+b²)/2)+(b²+c²)/2) /2 ]^1/2

which is =

(a²+b²+b²+c²) ...now i think that rearrangement inequality can be employed to prove the final answer ...

sandeep ramesh

Blazing goIITian

Joined: 13 Mar 2008

Posts: 1183

14 Mar 2008 19:30:09 IST

Like 0 people liked this

why are we ruffians sir? Becos we attack all problems? Mr. Green

anchit saini

Blazing goIITian

Joined: 1 Feb 2008

Posts: 1251

14 Mar 2008 19:34:46 IST

Like 1 people liked this

ruffian

noun

1. A rough, violent person who engages in destructive actions

just to add
those destructive actions are done on problems!!

Hari Shankar

Forum Expert

Joined: 28 Feb 2007

Posts: 2173

14 Mar 2008 19:41:09 IST

Like 0 people liked this

marauders is more like it

Quick Reply

Reply

	Some HTML allowed.
	Keep your comments above the belt or risk having them deleted.
	Signup for a avatar to have your pictures show up by your comment
	If Members see a thread that violates the Posting Rules, bring it to the attention of the Moderator Team

Free Sign Up!

Preparing for IIT-JEE ?

Arihant Revision Package for IIT JEE - Books, Practice Tests + Rank Predictor

@ INR 1,995/-

For Quick Info

Find Posts by Topics

Physics.

Topics

9389

20086

3671

2185

7614

2826

2634

Mathematics.

Chemistry.

Biology

Institutes

Parents

Board

Fun Zone