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Problem Of The Week #487 October 19th 2021

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anemone

MHB POTW Director
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Feb 14, 2012
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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,941
Congratulations to lfdahl again for his correct solution.
Suggested solution from other:
Without loss of generality, suppose $z\le \dfrac{1}{2}$. Thus we write $x+y=\dfrac{1}{2}+t,\,z=\dfrac{1}{2}-t$ where $-\le t \le \dfrac{1}{2}$.

It is easy to see that $xy+yz+zx-2xyz=xy(1-2z)+(x+y)z\ge 0$.

$\begin{align*}xy+yz+zx-2xyz&=xy(1-2z)+(x+y)z\\& \le \left(\dfrac{x+y}{2}\right)^2(1-2z)+(x+y)z\\&=2t\left(\dfrac{1}{4}+\dfrac{t}{2}\right)^2+\dfrac{1}{4}-t^2\\&=\dfrac{4t^3-4t^2+t-2}{8}\\&=f(t)\end{align*}$.

Let $f'(t)=0$ and we get $t=\dfrac{1}{6}$ and $t=\dfrac{1}{2}$.

$f''\left(\dfrac{1}{6}\right)=-7.5<0$ tells us the maximum of $f$ is $f\left(\dfrac{1}{6}\right)=\dfrac{7}{27}$.
 
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