# Need Help

Hi , I was trying out one of the IMO 1984 problems :

If $$x,y$$ and $$z$$ are non-negative real numbers such that $$x+y+z=1$$ , prove the following inequality

$xy + yz +zx -2xyz \le \frac{7}{27}$

My approach :

Using the identities:

$1. x^3 + y^3 + z^3 - 3xyz = (x+y+z)(x^2+y^2+z^2-xy-yz-zx)$

$2. (x+y+z)^2 = x^2+y^2+z^2+2xy+2yz+2zx$

The given inequality can be rewritten as :

$2(x^3+y^3+z^3) - \frac{3}{2}(x^2+y^2+z^2) +\frac{1}{2} \ge \frac{2}{9}$

Consider the following function:

$f(t) = 2t^3 - \frac{3}{2}t^2 + \frac{1}{6}$

The above function becomes convex for t $\ge \frac{1}{4}$

So, using Jensen's inequality on $a,b$ and $c$

$\frac{f(a)+f(b)+f(c)}{3} \ge f(\frac{1}{3})$

We get,

$2(x^3+y^3+z^3) - \frac{3}{2}(x^2+y^2+z^2) +\frac{1}{2} \ge 2/9$

So, I have proved the inequality . I want to confirm if my approach is correct. I think that it is not the complete proof as this proof included only those $x,y$ and $z$ which are greater than $\frac{1}{4}$ . I do not have the proof that an yet lower value cannot be achieved when i consider all the $x,y$ and $z$ which range from 0 to 1.

Note by A Former Brilliant Member
3 years, 5 months ago

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@Calvin Lin , @Chew-Seong Cheong, @Daniel Liu ,@Pi Han Goh can u please help me out with this .

- 3 years, 5 months ago

Try homogenizing the inequality.

- 3 years, 5 months ago