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Algebra

Blazing goIITian

Joined:	1 Feb 2008
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15 Mar 2008 19:18:14 IST

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1380

Challenge 4

None

Let a,b,c be non negative real numbers such that

a² + b²+ c² + abc = 4

Prove that --

0 <= ab + bc + ca - abc <= 2

(there are more than one solutions)

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sandeep ramesh

Blazing goIITian

Joined: 13 Mar 2008

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15 Mar 2008 22:01:35 IST

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how come you know the first one?

pardesi .svk

Scorching goIITian

Joined: 7 Dec 2007

Posts: 219

15 Mar 2008 23:30:02 IST

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this can be done using these subs.
$a=2p,b=2q,c=2r \Rightarrow \sum p^{2} +2pqr=1 \Rightarrow$
p,q,r can be $\cos A ,\cos B ,\cos C$ where A,B,C are angles of a triangle

sandeep ramesh

Blazing goIITian

Joined: 13 Mar 2008

Posts: 1183

15 Mar 2008 23:30:40 IST

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thalai welcome here! why no posting here?

anchit saini

Blazing goIITian

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16 Mar 2008 07:10:15 IST

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it would be really appreciated if some more people try this question.

rhlaggarwal

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Joined: 16 Mar 2008

Posts: 9

16 Mar 2008 07:30:24 IST

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a^2 + b^2 + c^2 > ab + bc +ca

implies ab+bc+ca - abc < = 4 - 2abc

But a^2+ b^2 + c^a > abc for abc < 4?

Hence ab+bc+ca<=2

Anand Hegde

Blazing goIITian

Joined: 12 Jun 2007

Posts: 1078

18 Mar 2008 15:02:14 IST

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source of the problem-USAMO 2001 Razz

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