Difference between revisions of "2015 IMO Problems/Problem 5"
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(1) Put <math>x=y=0</math> in the equation, | (1) Put <math>x=y=0</math> in the equation, | ||
− | We get<math> f(0 + f(0)) + f(0) = 0 + f(0) + 0 | + | We get<math> f(0 + f(0)) + f(0) = 0 + f(0) + 0</math> |
− | or f(f(0)) = 0</math> | + | or <math>f(f(0)) = 0</math> |
Let <math>f(0) = k</math>, then <math>f(k) = 0</math> | Let <math>f(0) = k</math>, then <math>f(k) = 0</math> | ||
Revision as of 13:04, 14 May 2017
Let be the set of real numbers. Determine all functions
:
satisfying the equation
for all real numbers and
.
Proposed by Dorlir Ahmeti, Albania
This problem needs a solution. If you have a solution for it, please help us out by adding it.
Let be the set of real numbers. Determine all functions
:
satisfying the equation
for all real numbers and
.
Proposed by Dorlir Ahmeti, Albania
This problem needs a solution. If you have a solution for it, please help us out by adding it.
(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
.