Permutation Math Olympiad Question (Challenging)

March’s Problem of the Month was a tough one on permutations. Only six people solved it! (Site: http://www.fen.bilkent.edu.tr/~cvmath/Problem/problem.htm)

The question goes as follows:

In each step one can choose two indices 1\leq k,l\leq 100 and transform the 100 tuple (a_1, \cdots, a_k, \cdots, a_l, \cdots, a_{100}) into the 100 tuple (a_1, \cdots, \frac{a_k}{2}, \cdots, a_l+\frac{a_k}{2}, \cdots, a_{100}) if a_k is an even number. We say that a permutation (a_1, \cdots, a_{100}) of (1, 2, \cdots, 100) is good if starting from (1,2,\cdots, 100) one can obtain it after finite number of steps. Find the total number of distinct good permutations of (1, 2, \cdots, 100).

The official solution is beautiful and uses induction.

 

Personally, I used a more brute force technique to get the same answer using equivalence class theory which I learnt in my first year of undergraduate math! It is not so bad in this question, since n is only 100, but for higher values of n the approach in the official solution would be better.

If you are looking for recommended Math Olympiad books, check out this page. In particular, if you are looking for more Math Olympiad challenges, do check out this book Mathematical Olympiad Challenges. In fact, any book by Titu Andreescu is highly recommended as he is the legendary IMO (International Math Olympiad) coach that led the USA team to a perfect score!

Advertisements

About mathtuition88

http://mathtuition88.com
This entry was posted in math problem of the month and tagged , , , . Bookmark the permalink.

3 Responses to Permutation Math Olympiad Question (Challenging)

  1. abyssbrain says:

    Haha, I would have also use the equivalence class theory to solve this question if I encountered it. But yeah, it would be ridiculously tedious to solve it if n is higher…

    I’ll have a closer look to their official solution, it seems interesting.

    Liked by 1 person

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s