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Blazing goIITian

Joined:	27 Dec 2010
Post:	1508

15 Jul 2011 08:21:13 IST

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prove that for any three real distinct numbers, a , b and c when a+b+c = 1 (1+a)(1+b)(1+c) > 8(1-a)

Engineering Entrance , JEE Main , JEE Advanced , Mathematics , Algebra

prove that for any three real distinct numbers, a , b and c when a+b+c = 1(1+a)(1+b)(1+c) > 8(1-a)(1-b)(1-c)

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jagdish singh

Blazing goIITian

Joined: 19 Jan 2008

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15 Jul 2011 11:02:10 IST

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Hemant is your Question is

jagdish singh

Blazing goIITian

Joined: 19 Jan 2008

Posts: 1070

16 Jul 2011 17:44:48 IST

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hemang

Blazing goIITian

Joined: 27 Dec 2010

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16 Jul 2011 21:08:51 IST

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i it did the hard way.
on evaluation and simplification we get,
2 + 9abc > 7ab + 7bc + 7ca.
so if we prove this we can prove the above thing.
we start from this (a+b)(a-b)² + (b+c)(b-c)² + (c+a)(c-a)² > 0

if we proceed now we can get there.

Hari Shankar

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Joined: 28 Feb 2007

Posts: 2185

17 Jul 2011 12:57:43 IST

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Rewrite the problem by substituting 1 by (a+b+c) as having to prove that

Setting and dividing by each factor on both sides by 2, we can further rewrite as

given x+y+z=1, to prove that which is well known and easy as

Multiplying the above three inequalities concludes the solution

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