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Problem Of The Week #387 Oct 9th, 2019

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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,894
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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Remember to read the POTW submission guidelines to find out how to submit your answers!
 
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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,894
Congratulations to the following members for their correct solution!(Cool)

1. kaliprasad
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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