MathDB

10

Part of 2022 Thailand Online MO

Problems(1)

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

4/4/2022
Let Q\mathbb{Q} be the set of rational numbers. Determine all functions f:Q→Qf : \mathbb{Q}\to\mathbb{Q} satisfying both of the following conditions.
[*] f(a)f(a) is not an integer for some rational number aa. [*] For any rational numbers xx and yy, both f(x+y)−f(x)−f(y)f(x + y) - f(x) - f(y) and f(xy)−f(x)f(y)f(xy) - f(x)f(y) are integers.
functionFunctional Equationsalgebranumber theoryfunctional equation