National BDMO 2016 : Junior 8

Discussion on Bangladesh Mathematical Olympiad (BdMO) National
dshasan
Posts: 66
Joined: Fri Aug 14, 2015 6:32 pm
Location: Dhaka,Bangladesh

National BDMO 2016 : Junior 8

Unread post by dshasan » Tue Jan 10, 2017 11:39 pm

In $\bigtriangleup ABC$ , $\angle A = 20$, $\angle B = 80$, $\angle C = 80$, $BC = 12$ units. Perpendicular $BP$ is drawn on $AC$ from from $B$ which intersects $AC$ at the point $P$. $Q$ is a point on $AB$ in such a way that $QB = 6$ units. Find the value of $\angle CPQ$.
The study of mathematics, like the Nile, begins in minuteness but ends in magnificence.

- Charles Caleb Colton

User avatar
ahmedittihad
Posts: 181
Joined: Mon Mar 28, 2016 6:21 pm

Re: National BDMO 2016 : Junior 8

Unread post by ahmedittihad » Wed Jan 11, 2017 4:55 pm

This was a nice problem. Let $C'$ be the reflection of $C$ w.r.t $BP$. Now, $\angle C'BC = 2*10$. So, $\angle QBC'=80-20=60$. As $QB=6, BC'=12$ and $\angle QBC' =60$ we see that $\triangle QBC'$ is a $30-60-90$ triangle. So, $\angle BQC'=90$. We get, $BQC'P$ is cyclic. So, $\angle CPQ= \angle BPC+\angle BPQ=\angle BPC+BC'Q=90+30=120$.
Q.E.D
Frankly, my dear, I don't give a damn.

User avatar
Kazi_Zareer
Posts: 86
Joined: Thu Aug 20, 2015 7:11 pm
Location: Malibagh,Dhaka-1217

Re: National BDMO 2016 : Junior 8

Unread post by Kazi_Zareer » Thu Jan 19, 2017 2:13 am

Solution:
Take $X,Y$ reflections of $B$ about $P,Q$ respectively and $Z$ reflection of $C$ about $P$.Now see that $\triangle ZYC$ is equilateral triangle, so $ZY=ZC=ZD$, $Z$ is circumcenter of $\triangle CXY$, thus $\angle CXY=30^\circ$, but $PQ$ is midline of $\triangle CXY$, so $PQ\parallel XY$, that's $\angle CPQ=120^\circ$.
We cannot solve our problems with the same thinking we used when we create them.

User avatar
Thamim Zahin
Posts: 98
Joined: Wed Aug 03, 2016 5:42 pm

Re: National BDMO 2016 : Junior 8

Unread post by Thamim Zahin » Thu Feb 02, 2017 7:11 pm

I had made the reflection but didn't get that it was a right triangle. Have to draw diagram as scale from now on.
I think we judge talent wrong. What do we see as talent? I think I have made the same mistake myself. We judge talent by the trophies on their showcases, the flamboyance the supremacy. We don't see things like determination, courage, discipline, temperament.

prottoy das
Posts: 17
Joined: Thu Feb 01, 2018 11:28 am
Location: Sylhet

Re: National BDMO 2016 : Junior 8

Unread post by prottoy das » Fri Feb 23, 2018 7:53 pm

[I'm solving this problem with the geometry of class 9-10.]
Take a point D in AC such that PD=CP. Join B and D.Now two triangle BCP and BPD are congruent.So BD=BC=12.Here angle PBQ=60.Draw a perpendicular from D to AB. Let it intersect AB at E.Now cos60=12/BE.Hence BE=6.But we know BQ=6.So E and Q are the same point.Now quadrilateral BPDQ is cyclic because the sum of the opposite angle of the quadrilateral is equal to 180.S0 angle BDQ=angle BPQ.In the right angled triangle BDQ=30.So BPQ=30 and angle CPQ=120

Post Reply