Problem Statement

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.

KolegaAI Solution

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: