MathDB

f(x+y)-f(x)-f(y) and f(xy)=f(x)f(y) are integers (f:Q--->Q)

Source: 2022 Thailand Online MO P10

April 4, 2022

functionFunctional Equationsalgebranumber theoryfunctional equation

Problem Statement

Let

\mathbb{Q}

be the set of rational numbers. Determine all functions

f : \mathbb{Q}\to\mathbb{Q}

satisfying both of the following conditions.
[*]

f(a)

is not an integer for some rational number

a

. [*] For any rational numbers

x

and

y

, both

f(x + y) - f(x) - f(y)

and

f(xy) - f(x)f(y)

are integers.