Inequality, triangle and cyclic sums : a problem for RMO and INMO training

Problem:

If a, b, c are the sides of a triangle, then prove that $2(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}) \geq \frac{a}{c}+ \frac{c}{b}+ \frac{b}{a}+3$

Solution:

“Reverse-engineering” this will not help !! One big hint is to think of cyclic expressions as most identities and inequaltties to triangles have such a nature.

Here you go:

Since a, b, c are the sides of a triangle, then the cyclic sum $\sum(a+c-b)(a-b)^{2} \geq 0$

But this sum is $\sum(a+c-b)(a^{2}-2ab+b^{2})=\sum(a^{3}+a^{2}c-a^{2}b-2a^{2}b-2abc+2ab^{2}+ab^{2}+b^{2}c-b^{3}) = \sum(4a^{2}c)-\sum(2a^{2}b)-6abc \geq 0$

By dividing this  by 2abc, we get $2(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}) \geq \frac{a}{c}+\frac{b}{a}+\frac{c}{b}+3$ exactly what we had to prove.

Ref: Problems for the Mathematical Olympiads (From the first team selection test to the IMO) by Andrei Negut.

More later,

Nalin Pithwa

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