# Problem Of The Week #387 Oct 9th, 2019

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#### anemone

##### MHB POTW Director
Staff member
Here is this week's POTW:

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If $x$ and $y$ are positive real numbers that satisfy the equation $x+4\sqrt{xy}-2\sqrt{x}-4\sqrt{y}+4y=3$, evaluate $\dfrac{\sqrt{x}+2\sqrt{y}+2014}{4-\sqrt{x}-2\sqrt{y}}$.

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#### anemone

##### MHB POTW Director
Staff member
Congratulations to the following members for their correct solution!

2. castor28
3. MegaMoh
4. lfdahl
5. Greg

Solution from Greg :
Note that the L.H.S. of the given equation factors as $(\sqrt x+2\sqrt y)(\sqrt x+2\sqrt y-2)$.

Setting $u=\sqrt x+2\sqrt y$ we obtain the quadratic $u^2-2u-3=(u-3)(u+1)=0$ and this implies that $\sqrt x+2\sqrt y=3$.

Thus we may conclude that $\frac{\sqrt{x}+2\sqrt{y}+2014}{4-\sqrt{x}-2\sqrt{y}}=2017$

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