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Algebra

Blazing goIITian

 Joined: 27 Dec 2010 Post: 1508
15 Jul 2011 08:21:13 IST
3 People liked this
4
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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 Main & 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)

Blazing goIITian

Joined: 19 Jan 2008
Posts: 1139
15 Jul 2011 11:02:10 IST
0 people liked this

Blazing goIITian

Joined: 19 Jan 2008
Posts: 1139
16 Jul 2011 17:44:48 IST
1 people liked this

Blazing goIITian

Joined: 27 Dec 2010
Posts: 1508
16 Jul 2011 21:08:51 IST
0 people liked this

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.

Forum Expert
Joined: 28 Feb 2007
Posts: 2217
17 Jul 2011 12:57:43 IST
2 people liked this

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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