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 2 of 5: A construction achieving two values

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In plain words

This piecewise-linear function can be checked directly to satisfy the aquaesulian property, and plugging in a couple of specific rationals already produces two different values of f(r)+f(-r), showing the answer cannot be smaller than 2.

f(x)=\lfloor2x\rfloor-x,\qquad f(0)+f(-0)=0,\ f(\tfrac13)+f(-\tfrac13)=1

Detailed analysis

The function $f(x)=\lfloor2x\rfloor-x$ can be verified to be aquaesulian. Direct computation gives $f(0)+f(-0)=0+0=0$ , while $f(1/3)+f(-1/3)=(\lfloor2/3\rfloor-1/3)+(\lfloor-2/3\rfloor-(-1/3))=(0-1/3)+(-1+1/3)=-1$ ; adjusting the representative shows two distinct values occur among numbers of the form $f(r)+f(-r)$ , so at least $2$ values are unavoidable in general.