## 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
Posts: 134
Joined: Tue Jan 18, 2011 1:31 pm

### Dhaka Higher Secondary 2011/10

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 $x + y + z$.

TIUrmi
Posts: 61
Joined: Tue Dec 07, 2010 12:13 am
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### Re: Dhaka Higher Secondary 2011/10

$\triangle ABC$ = $\triangle APC$ + $\triangle BPC$ + $\triangle BPA$ = $\frac {1}{2}$ $AC.1$ + $\frac{1}{2}.BC.2$ + $\frac{1}{2}AB.3$= $\frac{1}{2}AB.BC$
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
Posts: 25
Joined: Mon Dec 13, 2010 4:23 pm
Location: Dhaka

### Re: Dhaka Higher Secondary 2011/10

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

Avik Roy
Posts: 156
Joined: Tue Dec 07, 2010 2:07 am

### Re: Dhaka Higher Secondary 2011/10

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
Posts: 25
Joined: Mon Dec 13, 2010 4:23 pm
Location: Dhaka

### Re: Dhaka Higher Secondary 2011/10

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