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 5 |
All IMO Problems and Solutions |