2023 USAMO Problems/Problem 2

Problem 2

Let $\mathbb{R}^{+}$ be the set of positive real numbers. Find all functions $f:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ such that, for all $x, y \in \mathbb{R}^{+}$ , $f(xy + f(x)) = xf(y) + 2$

Solution

First, let us plug in some special points; specifically, plugging in $x=0$ and $x=1$ , respectively:

$\begin{align} f(f(0)) &= 2 \\ f(y + f(1)) &= f(y) + 2 \end{align}$

Next, let us find the first and second derivatives of this function. First, with (2), we isolate $f(y)$ one one side

$\begin{align*} f(y) = f(y + f(1)) - 2 \end{align*}$

and then take the derivative:

$\begin{align*} \dfrac{\mathrm{d}f}{\mathrm{d}y} &=\dfrac{\mathrm{d}f}{\mathrm{d}y}\left[f(y + f(1)) - 2\right] \\ &= \dfrac{\mathrm{d}f}{\mathrm{d}y}\left[f(y + f(1))\right] - \dfrac{\mathrm{d}f}{\mathrm{d}y}\left[2\right] \\ &= f'(y + f(1))\cdot\dfrac{\mathrm{d}f}{\mathrm{d}y}\left[y + f(1)\right] \\ &= f'(y + f(1))\cdot(1)\\ f'(y) &= f'(y + f(1))\\ \end{align*}$

The second derivative is as follows:

$\begin{align*} \dfrac{\mathrm{d}^2f}{\mathrm{d}y^2} &= \dfrac{\mathrm{d}f}{\mathrm{d}y}\left[\dfrac{\mathrm{d}f}{\mathrm{d}y}\right] \\ &= \dfrac{\mathrm{d}f}{\mathrm{d}y}\left[f'(y + f(1))\right] \\ &= f''(y + f(1))\cdot\dfrac{\mathrm{d}f}{\mathrm{d}y}\left[y + f(1)\right] \\ f''(y) &= f''(y + f(1))\\ \end{align*}$

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 $f'$ and $f''$ must be constants, and $f$ 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, $f$ and $x$ , making $f$ depend on $x$ , which is not a constant function. That means we can model $f(x)$ like so:

$\begin{align*} f(x) = ax + b \end{align*}$

Via (1), we get the following:

$\begin{align*} f(f(0)) &= 2 \\ a(a(0) + b) + b &= 2 \\ ab + b &= 2 \end{align*}$

And via (2),

$\begin{align*} f(y + f(1)) &= f(y) + 2 \\ a(y + a(1) + b) + b &= ay + b + 2 \\ ay + a^2 + ab + b &= ay + b + 2 \\ a^2 + ab &= 2 \\ \end{align*}$

Setting these equations equal to each other,

$\begin{align*} ab + b &= a^2 + ab \\ b &= a^2 \\ \end{align*}$

Therefore,

$\begin{align*} ab + b &= 2 \\ a^3 + a^2 &= 2 \\ \end{align*}$

There are three solutions to this equation: $a = 1$ , $a = -1 + i$ , and $a = -1 - i$ . Knowing that $b = a^2$ , the respective $b$ values are $b = 1$ , $b = -2i$ , and $b = 2i$ . Thus, $f(x)$ could be the following: