Let Q be the set of rational numbers. A function f : Q → Q is called aquaesulian if the following property holds: for every
$$ x, y ∈ Q, f(x + f(y)) = f(x) + y $$
or
$$ f(f(x) + y) = x + f(y) $$
Show that there exists an integer c such that for any aquaesulian function f there are at most c different rational numbers of the form f(r) + f(−r) for some rational number r, and find the smallest possible value of c.
We'll break this down into several steps, using the insights gained from our analysis.
Step 1: Analyzing the Aquaesulian Property
Let's start by examining the two conditions that define an aquaesulian function:
$$
f(x + f(y)) = f(x) + y
$$
$$
f(f(x) + y) = x + f(y)
$$
Lemma 1: For any aquaesulian function f and any rational number a, either
$$ f(f(a)) = a or f(a + f(a)) = f(a) + a. $$
Proof:
If condition 1 holds for x = a and y = a, then
$$ f(a + f(a)) = f(a) + a $$
If condition 2 holds for x = a and y = 0, then
$$ f(f(a) + 0) = a + f(0) $$
which simplifies to
$$ f(f(a)) = a $$