2012 IMO Problems/Problem 4
Find all functions such that, for all integers and that satisfy , the following equality holds: (Here denotes the set of integers.)
Solution
Consider Then Now we look at
We can write
If , then
Case 1:
Case 2: , we will have or
Case 2.1: if is odd, if is even.
Case 2.2:
or
Case 2.2.1:
and
or and or
Case 2.2.2:
or
and or
If then
We will prove by induction
If then is true for some .
and if the statement is true for
or
and or
the statement is true for as well.
As the statement is true for , by mathematical induction we can conclude
So, Case 2.1, Case 2.2.1 and Case 2.2.2 are the three independent possible solutions.
--Dineshram
See Also
2012 IMO (Problems) • Resources | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 5 |
All IMO Problems and Solutions |