- #1

FeDeX_LaTeX

Gold Member

- 437

- 13

## Homework Statement

Let a, b and c be the lengths of the sides of a triangle. Suppose that ab + bc + ca = 1. Show that (a+1)(b+1)(c+1) < 4.

## Homework Equations

N/A (Olympiad problem)

## The Attempt at a Solution

I've made a few inroads at this problem, but whether or not they're going to take me in the right direction, I'm not sure. I used the arithmetic/harmonic mean inequality to show that:

[tex]9abc \leq a + b + c[/tex]

and I've used substitutions from their original equation to arrive at

[tex]8(a+b+c) \leq (a+b)(1+9c^2) + (b+c)(1+9a^2) + (a+c)(1+9b^2)[/tex]

but I'm puzzled at where to go from here. I don't think it would be useful to consider specific cases such as the equilateral triangle (a = b = c), since we appear to be trying to prove a general case, and I know I mustn't start from their solution and show that it's equal to the original equation.

Am I taking the right approach? Is a geometric approach better? Any hints/tips anyone can give, without giving me the answer?

Thanks.