MathDB

A1

Part of 2015 Indonesia MO Shortlist

Problems(1)

f_1 (x) f_2 (x) is periodic when f_1 (x), f_2 (x) are periodic

Source: INAMO Shortlist 2015 A1

5/4/2019
Function f:R→Rf: R\to R is said periodic , if ff is not a constant function and there is a number real positive pp with the property of f(x)=f(x+p)f (x) = f (x + p) for every x∈Rx \in R. The smallest positive real number p which satisfies the condition f(x)=f(x+p)f (x) = f (x + p) for each x∈Rx \in R is named period of ff. Given aa and bb real positive numbers, show that there are periodic functions f1f_1 and f2f_2, with periods aa and bb respectively, so that f1(x)⋅f2(x)f_1 (x)\cdot f_2 (x) is also a periodic function.
algebrafunctionperiodicperiod