1989 IMO Problems/Problem 1
Problem
Prove that in the set can be expressed as the disjoint union of subsets such that
i.) each contains 17 elements
ii.) the sum of all the elements in each is the same.
Solution
Let us start pairing numbers in the following fashion, where each pair sums to :
There are a total of pairs above. Let us start putting these pairs into each of the subsets starting with the first pair going into , into and so on with going into .
Now we have numbers present in each of the subsets all of which have the same total sum. We need more number to be filled in each subset from the remaining numbers .
Let us arrange these numbers in the following manner:
Now notice that for each number that's off by from , there's a counter number off by from in the opposite direction.
So we need to create similar imbalances in the Subsets to , so that we could add the above numbers to those imbalances to keep the total sum unchanged.
Now start swapping the number of each pair that we added to subsets to to create the above imbalances:
Swap from with from - this creates an imbalance of and
Swap from with from - this creates an imbalance of and
Swap from with from - creates an imbalance of and
Leave in as it is - imbalance.
Similarly,
Swap from with from - this creates an imbalance of and
Swap from with from - this creates an imbalance of and
Swap from with from - this creates an imbalance of and
Now start adding the numbers from to the subsets to to counter the imbalances to so that the sum remains unchanged. Notice that each subset now has elements with total sum = .
See Also
1989 IMO (Problems) • Resources | ||
Preceded by First question |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All IMO Problems and Solutions |