2010 IMO Problems/Problem 5
Problem
Each of the six boxes ,
,
,
,
,
initially contains one coin. The following operations are allowed
Type 1) Choose a non-empty box ,
, remove one coin from
and add two coins to
;
Type 2) Choose a non-empty box ,
, remove one coin from
and swap the contents (maybe empty) of the boxes
and
.
Determine if there exists a finite sequence of operations of the allowed types, such that the five boxes ,
,
,
,
become empty, while box
contains exactly
coins.
Author: Unknown currently
Solution
Let the notation be the configuration in which the
-th box has
coin,
Let .
Our starting configuration is
We need these compound moves:
Compound move 1: , this is just repeated type 1 move on all
coins.
Compound move 2: , apply type 1 move on 1 of the coin to get
, then apply compound move 1 to the 2 coins to get
, apply type 2 move and get
, and repeat compound move 1 and type 2 move until
is achieve.
Compound move 3: , apply compound move 2 to obtain
and use type 2 move to get
Compound move 4: with
's. Apply compound move 3
times.
Compound move 5: with
, use type 2 move
times.
Let's follow this move:
Using Compound move 1, 4 and 5, We can obtain:
, where where X has
2's and Y has
2's, and
is clearly bigger then
See also
2010 IMO (Problems) • Resources | ||
Preceded by Problem 4 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 6 |
All IMO Problems and Solutions |