<s>[INST] Prove that $(x+y+z)^2(yz+zx+xy)^2 \leq 3(y^2+yz+z^2)(z^2+zx+x^2)(x^2+xy+y^2)$ This is Problem 6 of the 2007 Indian National Math Olympiad (INMO).

If $x, y, z$ are positive real numbers, prove that
$(x+y+z)^2(yz+zx+xy)^2 \leq 3(y^2+yz+z^2)(z^2+zx+x^2)(x^2+xy+y^2).$

My best idea was to expand this and simplify. Although that doesn't look very feasible. Another idea is to see that $x^2+y^2+xy \geq x^2+y^2$. Then we just have to show that $(x+y+z)^2(yz+zx+xy)^2 \leq 3(x^2+y^2)(x^2+z^2)(y^2+z^2)$ if that is even true.
 [/INST] Let I = 3(x² + xy + y²)(y² + yz + z²)(z² + zx + x²)
Then, By Holder's Inequality, it follows that
I >= 81x²y²z²
Thus, we need to prove that
81x²y²z² >= (x + y + z)²(xy + yz + zx)²
Or 9xyz >= (x + y + z)(xy + yz + zx)
Expanding R.H.S. gives
9xyz >= xy(x + y) + yz(y + z) + zx(z + x) + 3xyz
Or  xy(x + y) + yz(y + z) + zx(z + x) >= 6xyz
Which is evidently true by AM>=GM. QED
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