Difference between revisions of "2023 USAMO Problems/Problem 2"
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== Problem 2 == | == Problem 2 == | ||
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Let <math>\mathbb{R}^{+}</math> be the set of positive real numbers. Find all functions <math>f:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}</math> such that, for all <math>x, y \in \mathbb{R}^{+}</math>,<cmath>f(xy + f(x)) = xf(y) + 2</cmath> | Let <math>\mathbb{R}^{+}</math> be the set of positive real numbers. Find all functions <math>f:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}</math> such that, for all <math>x, y \in \mathbb{R}^{+}</math>,<cmath>f(xy + f(x)) = xf(y) + 2</cmath> | ||
Revision as of 11:25, 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. 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