Problem 6

A function

f:\mathbb Q\to\mathbb Q

is called aquaesulian if the following property holds: for every

x,y\in\mathbb Q

f(x+f(y))=f(x)+y

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

Step 4 of 5: A dichotomy from applying the property once more

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In plain words

Feeding a cleverly chosen pair into the defining property and using injectivity splits every situation into exactly two clean outcomes, one of which pins down f(r) plus f(-r) as zero and the other of which expresses it in terms of f(f(s)) minus s.

s\to r\implies f(r)+f(-r)=0\ \text{or}\ f(f(s))=s+f(r)+f(-r)

Detailed analysis

Given $s\to r$ , apply the hypothesis to $x=s+f(r)$ and $y=-r$ : one of $x\to y$ or $y\to x$ holds. Direct computation of both sides using $s\to r$ and injectivity of $f$ (Step 3) shows that if $x\to y$ holds, then $f(r)+f(-r)=0$ , while if $y\to x$ holds, then $f(f(s))=s+f(r)+f(-r)$ .