BdMO Higher secondary solutions

[TIUrmi]

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।

[Masum]

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.

[Masum]

It would be better if you post the problems.

[Moon]

Masum, I think that she would appreciate more if you give us a full solution, not just the answer.

[Moon]

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.

[TIUrmi]

Thank you!
And yes I would prefer a 'solution' not an answer.
Check the resources section of the official website for the questions/

[Moon]

আচ্ছা একটা ভাল কথা। এইখানে সবাই কি গাইড লাইনটা পড়ে দেখেছ? আর LaTeX font install করেছ?

[Masum]

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

[Zzzz]

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).

[Moon]

আলাদা পোস্ট কর! (এইটা এইখানে থাকুক সমস্যা নাই...খালি কপি-পেস্ট কর)