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 1 of 5: Notation and the always-true case x to x

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In plain words

Naming the relation lets the problem's hypothesis be read as "for every pair, at least one direction holds," and taking both variables equal to the same value always trivially satisfies the relation.

x\to y:\iff f(x+f(y))=f(x)+y

Detailed analysis

Write $x\to y$ to mean $f(x+f(y))=f(x)+y$ ; the problem's condition says that for all $x,y$ , at least one of $x\to y$ or $y\to x$ holds. Taking $y=x$ trivially gives $x\to x$ , i.e. $f(x+f(x))=x+f(x)$ for every $x$ .