Dhaka Higher Secondary 2011/10

Problem for Higher Secondary Group from Divisional Mathematical Olympiad will be solved here.
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Please don't post problems (by starting a topic) in the "Higher Secondary: Solved" forum. This forum is only for showcasing the problems for the convenience of the users. You can post the problems in the main Divisional Math Olympiad forum. Later we shall move that topic with proper formatting, and post in the resource section.
BdMO
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Dhaka Higher Secondary 2011/10

Unread post by BdMO » Fri Jan 28, 2011 10:35 pm

A point $P$ is chosen inside a right angled triangle $ABC$, perpendicular lines $PS$, $PQ$ and $PR$ are drawn from $P$ on $AB, BC$ and $AC$. $PR = 1, PQ = 2$ and $PS = 3$ and $\angle RPS = 150^{\circ} $. The length of $ AB$ can be written in the form $x \sqrt y +z$ where $x, y, z$ are integers. Find $ x + y + z$.

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TIUrmi
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Re: Dhaka Higher Secondary 2011/10

Unread post by TIUrmi » Sun Jan 30, 2011 12:23 pm

$\triangle ABC$ = $\triangle APC$ + $\triangle BPC$ + $\triangle BPA$ = $\frac {1}{2}$ $AC.1$ + $\frac{1}{2}.BC.2$ + $\frac{1}{2}AB.3$= $\frac{1}{2}AB.BC$
So, $AB$ =$\frac {AC}{BC}$ + 2$\frac {BC}{BC}$ + 3$\frac{AB}{AC}$

By simple angle chasing we will get $\angle ACB$ = 60
= sec 60 +2 + 3 tan 60
= 2 + 2 + 3 $\sqrt {3}$
= 3 $\sqrt {3}$ + 4

$x$ +$y$ + $z$ = $10$
"Go down deep enough into anything and you will find mathematics." ~Dean Schlicter

ibrahim
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Re: Dhaka Higher Secondary 2011/10

Unread post by ibrahim » Sat Sep 03, 2011 9:23 pm

I've got 8.
ABC=APC+BPC+BPA

1/2 AB*AC=1/2 AC*1+1/2*2*BC+1/2*3*AB

AB=1+ 2*BC/AC+3*AB/AC

=1+2cos60+3tan60

=1+1+3*(3)^(1/2)

so x+y+z=8


akashe shurjota othe nirobe

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Avik Roy
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Re: Dhaka Higher Secondary 2011/10

Unread post by Avik Roy » Sat Sep 03, 2011 10:34 pm

Ibrahim, it's expected that you use LATEX in this forum

And draw a diagram and you should have the explanation for why your answer is incorrect
"Je le vois, mais je ne le crois pas!" - Georg Ferdinand Ludwig Philipp Cantor

ibrahim
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Re: Dhaka Higher Secondary 2011/10

Unread post by ibrahim » Sat Sep 03, 2011 10:48 pm

I am using on mobile, so i can't using latex. Plz explain....

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