2001 C7

Discussion on International Mathematical Olympiad (IMO)
User avatar
ahmedittihad
Posts: 181
Joined: Mon Mar 28, 2016 6:21 pm

2001 C7

Unread post by ahmedittihad » Thu Jan 12, 2017 7:56 pm

A pile of $n$ pebbles is placed in a vertical column. This configuration is modified according to the following rules. A pebble can be moved if it is at the top of a column which contains at least two more pebbles than the column immediately to its right. (If there are no pebbles to the right, think of this as a column with 0 pebbles.) At each stage, choose a pebble from among those that can be moved (if there are any) and place it at the top of the column to its right. If no pebbles can be moved, the configuration is called a final configuration. For each $n$, show that, no matter what choices are made at each stage, the final configuration obtained is unique. Describe that configuration in terms of $n$.

(I don't know why it was made a C7. Quite easy)
Frankly, my dear, I don't give a damn.

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

Re: 2001 C7

Unread post by Thanic Nur Samin » Fri Jan 13, 2017 12:50 am

Hint 1: Every final config is decreasing and difference between consecutive terms is $0$ or $1$.

Hint 2: There is at most one term so that the term immediately next to it is equal to it.

Hint 3: What can one say about the difference between two consecutive triangular number?
Hammer with tact.

Because destroying everything mindlessly isn't cool enough.

Post Reply