Beginner's Marathon

For students of class 6-8 (age 12 to 14)
User avatar
Thanic Nur Samin
Posts: 176
Joined: Sun Dec 01, 2013 11:02 am

Re: Beginner's Marathon

Unread post by Thanic Nur Samin » Thu Mar 30, 2017 1:15 am

Epshita32 wrote:P7. There are 2000 points on a circle and each point is given a number which is equal to the average of the two numbers which are its nearest neighbors. Show that all the numbers must be equal.
An alternate Solution:

We use extremal principal. Since there a finite number of numbers, there must exists a minimum number. If the minimal number occurs more than once, randomly pick one. Due to minimality, the numbers on the both sides of those numbers would be equal to the minimal number. This way, all the numbers on the circle will be equal to the minimal number and thus we are done.
Hammer with tact.

Because destroying everything mindlessly isn't cool enough.

User avatar
Thanic Nur Samin
Posts: 176
Joined: Sun Dec 01, 2013 11:02 am

Re: Beginner's Marathon

Unread post by Thanic Nur Samin » Thu Mar 30, 2017 1:21 am

Problem 8:

Suppose that 5 points lie on a sphere. Prove that there exists a closed semi-sphere (half a sphere including boundary), which contains 4 of the points.
Hammer with tact.

Because destroying everything mindlessly isn't cool enough.

Epshita32
Posts: 37
Joined: Mon Aug 24, 2015 12:34 am

Re: Beginner's Marathon

Unread post by Epshita32 » Thu Mar 30, 2017 9:09 pm

Solution to P8:
Consider the circle through any two of the points. This partitions the sphere into two hemispheres. By PHP, 2 of the remaining 3 points must lie in one of the hemispheres. These two points and the original two points lie in a closed semi-sphere.

User avatar
ahmedittihad
Posts: 172
Joined: Mon Mar 28, 2016 6:21 pm

Re: Beginner's Marathon

Unread post by ahmedittihad » Thu Mar 30, 2017 9:21 pm

$\text{Problem }9$

There is a billiard ball rolling on a circular table. Everytime it hits the edge, it gets reflected (Assume the ball hits the circle at X . The course of the ball gets reflected with respect to the tangent from X). Prove that, if the ball goes through a point $P$ three times, the ball goes through that point $P$ infinitely many times. Provided that the ball doesn't stop.
Frankly, my dear, I don't give a damn.

User avatar
Ananya Promi
Posts: 18
Joined: Sun Jan 10, 2016 4:07 pm
Location: Naogaon, Bangladesh

Re: Beginner's Marathon

Unread post by Ananya Promi » Fri Mar 31, 2017 6:28 pm

Another solution to P6:
As $EM$ is perpendicular on $AC$ we can say by perpendicular lemma,
$CE^2 + AM^2 = AE^2 + CM^2$
or, $CM^2 - AM^2 = CE^2 - AE^2$
Again, $MF$ is perpendicular on $BC$
so, $CF^2 + BM^2 = CM^2 + BF^2$
or, $CM^2 - BM^2 = CF^2 - BF^2 = CM^2 - AM^2$
so, $CE^2 - AE^2 = CF^2 - BF^2$
or, $CE^2 + BF^2 = CF^2 + AE^2$
or, $CE^2 + BF^2 + BM^2 = CF^2 + AE^2 + AM^2$
or, $CE^2 + MF^2 = CF^2 + EM^2$
So, by the reverse of perpendicular lemma, we can say,
$CM$ is perpendicular on $EF$
So, $EAMD$ is cyclic.
$\angle{EMA} = \angle{ADE}$
Again, $MDFB$ is cyclic.
$\angle{BMF} = \angle{BDF}$
So, $\angle{EMF} = \angle{ADB} = 180^o - \angle{EMA} - \angle{BMF} = 180^o - \angle{ADE} - \angle{BDF}$
Proved

User avatar
Zawadx
Posts: 89
Joined: Fri Dec 28, 2012 8:35 pm

Re: Beginner's Marathon

Unread post by Zawadx » Fri Mar 31, 2017 8:25 pm

So got a bit out of hand. Reducing the max difficulty level to the hardest problems in BDMO. Please don't post problems which ask for theory beyond that presented in the book. If you use advanced ideas, try to explain them so that everyone will understand :)

Also, let's try to keep this on track with a proper progression of problems and solutions, so that readers can work through them from the start. If you want to post an alternate solution but other problems have already been posted, I recommend starting a new thread. They're very welcome! Also if you must make a comment, please use [*hide] tags so that it doesn't clutter the page. Though if you want to highlight why a solution is wrong (and preferably post the correct solution), do it outside the [*hide] tag.

If you want to catch my attention about a too difficult problem, a wrong solution or anyone trolling, please use the Report button. It's the red exclamation button just beside "QUOTE. This will ensure I can fix the problem as soon as possible.

User avatar
ahmedittihad
Posts: 172
Joined: Mon Mar 28, 2016 6:21 pm

Re: Beginner's Marathon

Unread post by ahmedittihad » Fri Apr 07, 2017 2:06 pm

ahmedittihad wrote:$\text{Problem }9$

There is a billiard ball rolling on a circular table. Everytime it hits the edge, it gets reflected (Assume the ball hits the circle at X . The course of the ball gets reflected with respect to the tangent from X). Prove that, if the ball goes through a point $P$ three times, the ball goes through that point $P$ infinitely many times. Provided that the ball doesn't stop.
This problem is really cool if you get the gist of it. I'll give some hints.
What happens if a line gets overlapped?(i,e the line is same to a previous one)
Can you find some invariance regarding the length of each line segment?(Here a line segment means the trajectory of the ball after getting reflected and before getting reflected again)
.
Frankly, my dear, I don't give a damn.

User avatar
Atonu Roy Chowdhury
Posts: 40
Joined: Fri Aug 05, 2016 7:57 pm
Location: Chittagong, Bangladesh

Re: Beginner's Marathon

Unread post by Atonu Roy Chowdhury » Fri Apr 07, 2017 9:34 pm

ahmedittihad wrote:$\text{Problem }9$

There is a billiard ball rolling on a circular table. Everytime it hits the edge, it gets reflected (Assume the ball hits the circle at X . The course of the ball gets reflected with respect to the tangent from X). Prove that, if the ball goes through a point $P$ three times, the ball goes through that point $P$ infinitely many times. Provided that the ball doesn't stop.
Here's my solution:
Lemma 1: All lengths of the chords of the ball's path is equal.
Proof: Let $AB$ and $BC$ be two consecutive chords of the ball's path and $O$ be the center of the circle. So, $ \angle ABO = \angle CBO$. So, $AB=BC$.
Lemma 2: There's exactly two chords of same length passing through a point $P$.
Proof: Let $AB$ and $CD$ be two chords of length $x$ passing through $P$. Let $PA = a$ and $PD=b$. So, by POP we get $a(x-a) = b(x-b)$ which implies $a=b$ or $a+b=x$. That means $P$ divides these two chords in two segments having a length of $a$ and $x-a$. Now, if there is three chords through point $P$, the circle $\circ (P,PA)$ intersects our main circle at three points. Contradiction!

Now, if the ball passes through $P$ three times, it will pass through segment $AB$ at least two times.
The rest is trivial.
Most of my solutions in BdMO forum ends with "The rest is trivial". :P

Epshita32
Posts: 37
Joined: Mon Aug 24, 2015 12:34 am

Re: Beginner's Marathon

Unread post by Epshita32 » Sat Apr 08, 2017 11:01 am

Problem 10.
Let $S$ be a square of side $2$, and choose $9$ points inside $S$. Show that $3$ of these points may be chosen which are the vertices of a triangle of area $\leq \dfrac{1}{2}$.

Btw, this is a really easy problem and please participate, everyone.
Last edited by Zawadx on Sat Apr 08, 2017 3:33 pm, edited 1 time in total.
Reason: Fixed LaTex

Absur Khan Siam
Posts: 61
Joined: Tue Dec 08, 2015 4:25 pm
Location: Bashaboo , Dhaka

Re: Beginner's Marathon

Unread post by Absur Khan Siam » Sun Apr 09, 2017 4:45 pm

Solution to problem $\boxed{10}$:
We partition the square in $4$ squares with the length of the side $1$.According to PHP, there is a square which contains at least $3$ of the points .We can see that the area of this triangle is smaller than the area of a triangle with the vertices on the sides of the square and then notice that the area of this second triangle is smaller
than the area of a triangle with the vertices in the vertices of the square. But the area of such a triangle is $\dfrac{1}{2}$.
Thus we proof that , area of that triangle is $\leq \dfrac{1}{2}$
"(To Ptolemy I) There is no 'royal road' to geometry." - Euclid

Post Reply