BdMO Higher secondary solutions

Discussion on Bangladesh Mathematical Olympiad (BdMO) National
User avatar
TIUrmi
Posts: 61
Joined: Tue Dec 07, 2010 12:13 am
Location: Dinajpur, Bangladesh
Contact:

BdMO Higher secondary solutions

Unread post by TIUrmi » Tue Dec 07, 2010 6:19 pm

I need the solutions of the latest National BdMO's last three questions of higher secondary catergory.

YaY!! I am the first one to post in the forum.
ইতিহাসে আমার নাম স্বর্ণাক্ষরে লেখা থাকবে সেই আনন্দে আমার কান্না পাচ্ছে। :cry:
"Go down deep enough into anything and you will find mathematics." ~Dean Schlicter

User avatar
Masum
Posts: 592
Joined: Tue Dec 07, 2010 1:12 pm
Location: Dhaka,Bangladesh

Re: BdMO Higher secondary solutions

Unread post by Masum » Tue Dec 07, 2010 7:29 pm

Well,I am posting the solution to a problem(may be 9 or 10).
The problem asked to find the number of odd coefficient in the expansion of $(a+b)^{2010}$.I am posting for exponent $n$.The number of odd coefficients is $2^m$ where $m$ is the number of $1's$ in the binary expansion of $n$.The rest is yours.
One one thing is neutral in the universe, that is $0$.

User avatar
Masum
Posts: 592
Joined: Tue Dec 07, 2010 1:12 pm
Location: Dhaka,Bangladesh

Re: BdMO Higher secondary solutions

Unread post by Masum » Tue Dec 07, 2010 8:14 pm

It would be better if you post the problems.
One one thing is neutral in the universe, that is $0$.

User avatar
Moon
Site Admin
Posts: 751
Joined: Tue Nov 02, 2010 7:52 pm
Location: Dhaka, Bangladesh
Contact:

Re: BdMO Higher secondary solutions

Unread post by Moon » Tue Dec 07, 2010 9:03 pm

Masum, I think that she would appreciate more if you give us a full solution, not just the answer. :)
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin

Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.

User avatar
Moon
Site Admin
Posts: 751
Joined: Tue Nov 02, 2010 7:52 pm
Location: Dhaka, Bangladesh
Contact:

Re: BdMO Higher secondary solutions

Unread post by Moon » Tue Dec 07, 2010 9:53 pm

This is the Problem: BdMO 2010, Higher Sec. ,problem 9.

Find the number of odd coefficients in expansion of $(x + y)^{2010}$.

I solve the problem using Lucus Theorem. Lucus Theorem actually KILLS the problem. Abir vi showed me another "elementary" way. I think that Avik vi can show us that solution, as he proposed this problem.
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin

Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.

User avatar
TIUrmi
Posts: 61
Joined: Tue Dec 07, 2010 12:13 am
Location: Dinajpur, Bangladesh
Contact:

Re: BdMO Higher secondary solutions

Unread post by TIUrmi » Wed Dec 08, 2010 1:12 am

Thank you!
And yes I would prefer a 'solution' not an answer. :)
Check the resources section of the official website for the questions/
"Go down deep enough into anything and you will find mathematics." ~Dean Schlicter

User avatar
Moon
Site Admin
Posts: 751
Joined: Tue Nov 02, 2010 7:52 pm
Location: Dhaka, Bangladesh
Contact:

Re: BdMO Higher secondary solutions

Unread post by Moon » Wed Dec 08, 2010 2:34 am

আচ্ছা একটা ভাল কথা। এইখানে সবাই কি গাইড লাইনটা পড়ে দেখেছ? আর LaTeX font install করেছ?
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin

Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.

User avatar
Masum
Posts: 592
Joined: Tue Dec 07, 2010 1:12 pm
Location: Dhaka,Bangladesh

Re: BdMO Higher secondary solutions

Unread post by Masum » Thu Dec 09, 2010 12:42 am

Moon wrote:This is the Problem: BdMO 2010, Higher Sec. ,problem 9.

Find the number of odd coefficients in expansion of $(x + y)^{2010}$.

I solve the problem using Lucus Theorem. Lucus Theorem actually KILLS the problem. Abir vi showed me another "elementary" way. I think that Avik vi can show us that solution, as he proposed this problem.
If you look carefully,you can find it as a consequence of Lucas-Kummer theorem.:) But my mistake was I assumed TIURMI knows this.And thanks to Moon for giving the link,also a general suggestion for users.If you see a new theory then search in google.Because that will help you so much that you will feel it yourself,for example Tricky lemma or Vieta Jumping
One one thing is neutral in the universe, that is $0$.

User avatar
Zzzz
Posts: 172
Joined: Tue Dec 07, 2010 6:28 am
Location: 22° 48' 0" N / 89° 33' 0" E

Re: BdMO Higher secondary solutions

Unread post by Zzzz » Thu Dec 09, 2010 1:37 pm

আচ্ছা, ৮,৯ নম্বরের সমাধান আগেও দেখছি। কিন্তু আমি কখনোই ১০ নম্বর সমস্যাটার সমাধান কোথাও দেখি নাই। নিজেও করতে পারি নাই অবশ্যই। কেউ কি দিতে পারেন?

সমস্যাটা হইলঃ

Let $a_1,a_2,…,a_k,…,a_n$ is a sequence of distinct positive real numbers such that $a_1<a_2<…<a_k$ and $a_k>a_{k+1}>…>a_n$. A grasshopper is to jump along the real axis, starting at the point $O$ and making $n$ jumps to right of lengths $a_1,a_2,…,a_n$ respectively. Prove that, once he reaches the rightmost point, he can come back to point $O$ by making $n$ jumps to left of of lengths $a_1,a_2,…,a_n$ in some order such that he never lands on a point which he already visited while jumping to the right. (The only exceptions are point O and the rightmost point).
Every logical solution to a problem has its own beauty.
(Important: Please make sure that you have read about the Rules, Posting Permissions and Forum Language)

User avatar
Moon
Site Admin
Posts: 751
Joined: Tue Nov 02, 2010 7:52 pm
Location: Dhaka, Bangladesh
Contact:

Re: BdMO Higher secondary solutions

Unread post by Moon » Thu Dec 09, 2010 1:47 pm

আলাদা পোস্ট কর! :) (এইটা এইখানে থাকুক সমস্যা নাই...খালি কপি-পেস্ট কর)
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin

Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.

Post Reply