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Saturday, January 16, 2016

IMO Inequality Problem

Let a,b,c be real numbers greater than 2 such that 1a+1b+1c=1.

Prove that (a2)(b2)(c2)1.

My solution:

Note that

(a2)(b2)(c2)

=abc(a2a)(b2b)(c2c)

=abc(12a)(12b)(12c)

=abc(1a+1b+1c2a)(1a+1b+1c2b)(1a+1b+1c2c)

=abc(1b+1c1a)(1a+1c1b)(1a+1b1c)

We now use the famous identity that says for all real and positive x,y and z, we have

xyz(x+yz)(x+zy)(y+zx)

In our case, we have x=1a,y=1b,z=1c and so we get

1abc(1a+1b1c)(1a+1c1b)(1b+1c1a)

i.e.

abc(1a+1b1c)(1a+1c1b)(1b+1c1a)1

The proof is then follows.

Equality occurs when a=b=c=3.


2 comments:

  1. Aww...thank you Michelle for your nice compliment, I appreciate it!

    ReplyDelete