2015 IMO Problems/Problem 5
Problem
Let be the set of real numbers. Determine all functions : satisfying the equation
for all real numbers and .
Proposed by Dorlir Ahmeti, Albania
Solution
for all real numbers and .
(1) Put in the equation, We get or Let , then
(2) Put in the equation, We get But so, or Hence
Case :
Put in the equation, We get or, Say , we get
So, is a solution
Case : Again put in the equation, We get or,
We observe that must be a polynomial of power as any other power (for that matter, any other function) will make the and of different powers and will not have any non-trivial solutions.
Also, if we put in the above equation we get
satisfies both the above.
Hence, the solutions are and .
See Also
2015 IMO (Problems) • Resources | ||
Preceded by Problem 4 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 6 |
All IMO Problems and Solutions |