Difference between revisions of "2012 IMO Problems/Problem 4"
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'''So, Case 2.1, Case 2.2.1 and Case 2.2.2 are the three independent possible solutions.''' | '''So, Case 2.1, Case 2.2.1 and Case 2.2.2 are the three independent possible solutions.''' | ||
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--Dineshram | --Dineshram | ||
==See Also== | ==See Also== | ||
*[[IMO Problems and Solutions]] | *[[IMO Problems and Solutions]] |
Revision as of 06:02, 4 December 2013
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