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]] | ||
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+ | {{IMO box|year=2012|num-b=3|num-a=5}} | ||
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+ | [[Category:Olympiad Algebra Problems]] | ||
+ | [[Category:Functional Equation Problems]] |
Latest revision as of 08:08, 20 November 2023
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 |