1994 IMO Problems/Problem 5
Problem
Let be the set of real numbers strictly greater than . Find all functions satisfying the two conditions:
1. for all and in ;
2. is strictly increasing on each of the intervals and .
Solution
The only solution is
Setting we get
Therefore, for
Note: If we can show that is always we will get that for all in and therefore,
Let Setting we get
If we have as well.
Consider We get Since is strictly increasing for and in this domain, we must have but since and we also have that which is a contradiction. Therefore
Consider Using a similar argument, we will get that but also which is a contradiction.
Hence, must be Since we can conclude that and therefore, for all .
See Also
1994 IMO (Problems) • Resources | ||
Preceded by Problem 4 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 6 |
All IMO Problems and Solutions |