Difference between revisions of "2012 IMO Problems/Problem 4"
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<math>\Rightarrow f(x) = f(1)x^2</math> <math>\forall</math> <math>x.\Box</math> | <math>\Rightarrow f(x) = f(1)x^2</math> <math>\forall</math> <math>x.\Box</math> | ||
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+ | --Dineshram | ||
==See Also== | ==See Also== | ||
*[[IMO Problems and Solutions]] | *[[IMO Problems and Solutions]] |
Revision as of 06:32, 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.
--Dineshram