<s>[INST] IMO 1984: Prove that $0 ≤ yz + zx +xy −2xyz ≤ \frac {7}{27}$, where $x,y$ and $z$ are non-negative real numbers for which $x + y + z = 1.$ I tried to solve this inequality because I find exotic.  Actually, I didn't look at the right solution.  Because before I look at the right solution, I want to know if my solution is right or not.

Prove that $0 ≤ yz + zx +xy -2xyz≤\frac{7}{27}$, where $x,y$ and $z$ are non-negative real numbers for which $x + y + z = 1.$

Attempts:
If $x=0$. The left side of the inequality is correct. So, I can accept $x,y,z≠0$.
It is enough to prove $\frac 1x+\frac1y+\frac1z≥2$
We have,

$x+y+z≥3\sqrt[3]{xyz}\Longrightarrow xyz≤\frac1{27}$
$\frac 1x+\frac1y+\frac1z≥\frac3{\sqrt[3]{xyz}}≥9≥2$
The left side proved.

It is obvious at least one of the numbers is less than $\frac 12.$

So, we can choose $y$, such that $y≤\frac 12$.

Therefore, we have

$yz + zx +xy −2xyz ≤ \frac7{27}$
$x(y+z)+yz(1-2x)-\frac7{27}≤0$
$x(1-x)+y(1-x-y)(1-2x)-\frac7{27}≤0$
$x-x^2+y-xy-y^2-2xy+2x^2y+2xy^2-\frac7{27}≤0$
$x^2-x-y+xy+y^2+2xy-2x^2y-2xy^2+\frac{7}{27}≥0$
$x^2(1-2y)+x(3y-2y^2-1)+(y^2-y+\frac7{27})≥0$
$2(\frac12-y)\left(x+\frac{y-1}{2}\right)^2+\frac{1}{108} (3y-1)^2(6y+1)≥0$

Of course, I'm not sure the solution is correct.  Can you verify the solution?
Thank you.
 [/INST] Assume $x= \max\{\,x,\,y,\,z\,\}\,\therefore\,3\,x\geqq 1$.
For $z= 1- x- y$, we need to prove
$$x^{\,2}+ y^{\,2}+ 3\,xy- 2\,x^{\,2}y- 2\,y^{\,2}x- x- y+ \frac{7}{27}\geqq 0$$
We have
$$\because\,x^{\,2}+ y^{\,2}+ 3\,xy- 2\,x^{\,2}y- 2\,y^{\,2}x- x- y+ \frac{7}{27}=$$ 
$$= \left \{ 4(\,1- 2\,x\,)y^{\,2}- 4(\,x- 1\,)(\,2\,x- 1\,)y+ \frac{1}{27}(\,5- 9\,x\,)^{\,2} \right \}+$$
$$+ \frac{1}{3}(\,2\,x- 1\,)(\,3\,y- 1\,)(\,3\,x+ 3\,y- 2\,)$$
$$\because\,x^{\,2}+ y^{\,2}+ 3\,xy- 2\,x^{\,2}y- 2\,y^{\,2}x- x- y+ \frac{7}{27}=$$
$$= \frac{1}{27}(\,3\,x- 1\,)^{\,2}- \frac{1}{9}(\,2\,x- 1\,)(\,3\,y- 1\,)(\,3\,x+ 3\,y- 2\,)$$
$$\because\,x^{\,2}+ y^{\,2}+ 3\,xy- 2\,x^{\,2}y- 2\,y^{\,2}x- x- y+ \frac{7}{27}=$$
$$= \frac{1}{108}(\,1- 3\,x\,)^{\,2}(\,6\,x+ 1\,)+ \frac{1}{4}(\,1- 2\,x\,)(\,x+ 2\,y- 1\,)^{\,2}$$
Because
$$4(\,1- 2\,x\,)y^{\,2}- 4(\,x- 1\,)(\,2\,x- 1\,)y+ \frac{1}{27}(\,5- 9\,x\,)^{\,2}\geqq 0$$
$$\because\,{\rm discriminant}[\,4(\,1- 2\,x\,)y^{\,2}- 4(\,x- 1\,)(\,2\,x- 1\,)y+ \frac{1}{27}(\,5- 9\,x\,)^{\,2},\,y\,]=$$ 
$$= \frac{32}{27}(\,2\,x- 1\,)(\,3\,x- 1\,)^{\,3}\leqq 0$$
So
$$x^{\,2}+ y^{\,2}+ 3\,xy- 2\,x^{\,2}y- 2\,y^{\,2}x- x- y+ \frac{7}{27}\geqq 0$$
$\lceil$ ANOTHER!way $\rfloor$
As @Elvin's above, we can choose $1- 2\,x\geqq 0$ (or $\because\,(\,1- 2\,x\,)(\,1- 2\,y)\leqq 0\,\because\,x+ y< 1$).
$$\because\,x^{\,2}+ y^{\,2}+ 3\,xy- 2\,x^{\,2}y- 2\,y^{\,2}x- x- y+ \frac{7}{27}=$$
$$= \frac{1}{108}(\,1- 3\,x\,)^{\,2}(\,6\,x+ 1\,)+ \frac{1}{4}(\,1- 2\,x\,)(\,x+ 2\,y- 1\,)^{\,2}\geqq 0$$
</s>