Baltic Way 1995 · Problem 12
Combinatorics
Assume we have 95 boxes and 19 balls distributed in these boxes in an arbitrary manner. We take six new balls at a time and place them in six of the boxes, one ball in each of the six. Can we, by repeating this process a suitable number of times, achieve a situation in which each of the 95 boxes contains an equal number of balls?
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Review
Topics
Games and strategies · Algorithms and processes · Induction and recursion
Solutions
Solution
Solution:
Since , we can put times balls in the boxes so that the number of balls in one of the boxes increases by two, while in all other boxes it increases by one. Repeating this procedure, we can either diminish the difference between the number of balls in the box which has most balls and the number of balls in the box with the least number of balls, or diminish the number of boxes having the least number of balls, until all boxes have the same number of balls.
Contest context
Results from Baltic Way 1995
9 teams
- Mean score
- 3.9 / 5
- Scores of 4 or 5
- 6 / 9
- Estonia
- 4 / 5
Score distribution
All team scores
| Team | Score |
|---|---|
| Poland | 5 / 5 |
| Latvia | 5 / 5 |
| Sweden | 5 / 5 |
| Lithuania | 3 / 5 |
| Denmark | 4 / 5 |
| Finland | 5 / 5 |
| St. Petersburg | 1 / 5 |
| Estonia | 4 / 5 |
| Iceland | 3 / 5 |