Difference between revisions of "2023 USAMO Problems/Problem 2"
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− | For both of these derivatives, we see that the input to the function does not matter: it will return the same result regardless of input. Therefore, the functions <math>f'</math> and <math>f''</math> must be constants, and <math>f</math> must be a linear equation. That means we can model <math>f(x)</math> like so: | + | For both of these derivatives, we see that the input to the function does not matter: it will return the same result regardless of input. Therefore, the functions <math>f'</math> and <math>f''</math> must be constants, and <math>f</math> must be a linear equation or a constant. We know it is not a constant because if it was, the problem could be reduced to a linear equation with two unknowns, <math>f</math> and <math>x</math>, making <math>f</math> depend on <math>x</math>, which is not a constant function. That means we can model <math>f(x)</math> like so: |
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− | ~ cogsandsquigs | + | ~cogsandsquigs |
Revision as of 11:33, 7 April 2023
Problem 2
Let be the set of positive real numbers. Find all functions
such that, for all
,
Solution
First, let us plug in some special points; specifically, plugging in and
, respectively:
Next, let us find the first and second derivatives of this function. First, with (2), we isolate one one side
and then take the derivative:
The second derivative is as follows:
For both of these derivatives, we see that the input to the function does not matter: it will return the same result regardless of input. Therefore, the functions and
must be constants, and
must be a linear equation or a constant. We know it is not a constant because if it was, the problem could be reduced to a linear equation with two unknowns,
and
, making
depend on
, which is not a constant function. That means we can model
like so:
Via (1), we get the following:
And via (2),
Setting these equations equal to each other,
Therefore,
There are three solutions to this equation: ,
, and
. Knowing that
, the respective
values are
,
, and
. Thus,
could be the following:
~cogsandsquigs