## BdMO Higher secondary solutions

TIUrmi
Posts: 61
Joined: Tue Dec 07, 2010 12:13 am
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### BdMO Higher secondary solutions

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।
"Go down deep enough into anything and you will find mathematics." ~Dean Schlicter

Masum
Posts: 592
Joined: Tue Dec 07, 2010 1:12 pm

### Re: BdMO Higher secondary solutions

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\$.

Masum
Posts: 592
Joined: Tue Dec 07, 2010 1:12 pm

### Re: BdMO Higher secondary solutions

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

Moon
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Joined: Tue Nov 02, 2010 7:52 pm
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### Re: BdMO Higher secondary solutions

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

learn how to write equations, and don't forget to read Forum Guide and Rules.

Moon
Posts: 751
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### Re: BdMO Higher secondary solutions

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

learn how to write equations, and don't forget to read Forum Guide and Rules.

TIUrmi
Posts: 61
Joined: Tue Dec 07, 2010 12:13 am
Contact:

### Re: BdMO Higher secondary solutions

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

Moon
Posts: 751
Joined: Tue Nov 02, 2010 7:52 pm
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### Re: BdMO Higher secondary solutions

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

learn how to write equations, and don't forget to read Forum Guide and Rules.

Masum
Posts: 592
Joined: Tue Dec 07, 2010 1:12 pm

### Re: BdMO Higher secondary solutions

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\$.

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

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.

Moon