BdMO National 2012: Junior 7, Primary 9

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Zzzz
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BdMO National 2012: Junior 7, Primary 9

Unread post by Zzzz » Sun Feb 12, 2012 9:22 am

Problem:
Each room of the Magic Castle has exactly one door. The rooms are designed such that when you can go from one room to the next one through a door, the second room's length is equal to the first room's width, and the second room's width is half of the first room's width (see the figure). Each door can be used only once. Magic Prince has entered the castle and now needs to get out. To get out of each room, the prince needs time equal to the width of the room. The prince has to use each door to get out of the castle. Due to the blessings of a Sufi, the prince can become as small as he wants (so that he can go into even very small rooms). If the castle is a square of side length $20$ meters then how long will it take for the prince to get out?
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SadidT7
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Re: BdMO National 2012: Junior 7, Primary 9

Unread post by SadidT7 » Wed Sep 02, 2015 7:56 pm

40 minutes.

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samiul_samin
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Re: BdMO National 2012: Junior 7, Primary 9

Unread post by samiul_samin » Wed Feb 21, 2018 8:53 pm

SadidT7 wrote:
Wed Sep 02, 2015 7:56 pm
40 minutes.
The answer is $\fbox {40 unit time}$

Solution
$10+10+5+5+2.5+2.5+1.25+1.25+...$
$\Rightarrow 20+10+5+2.5+...$
So,in this infinite sequence,
First term $a=20$
Rate of change $r=\dfrac {10}{20}=\dfrac 12$
Total time
$S=\dfrac {a}{1-r}$

$\Rightarrow S=\dfrac {20}{1-\dfrac 12}=20×2=40$.

The answer is $\fbox {40 unit time}$

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