MathDB

28

Part of 1978 IMO Longlists

Problems(1)

Two functions and their properties.

Source:

10/28/2010
Let c,sc, s be real functions defined on R∖{0}\mathbb{R}\setminus\{0\} that are nonconstant on any interval and satisfy c(xy)=c(x)c(y)−s(x)s(y) for any x≠0,y≠0c\left(\frac{x}{y}\right)= c(x)c(y) - s(x)s(y)\text{ for any }x \neq 0, y \neq 0 Prove that: (a)c(1x)=c(x),s(1x)=−s(x)(a) c\left(\frac{1}{x}\right) = c(x), s\left(\frac{1}{x}\right) = -s(x) for any x=0x = 0, and also c(1)=1,s(1)=s(−1)=0c(1) = 1, s(1) = s(-1) = 0; (b)c(b) c and ss are either both even or both odd functions (a function ff is even if f(x)=f(−x)f(x) = f(-x) for all xx, and odd if f(x)=−f(−x)f(x) = -f(-x) for all xx). Find functions c,sc, s that also satisfy c(x)+s(x)=xnc(x) + s(x) = x^n for all xx, where nn is a given positive integer.
functionlogarithmsalgebra unsolvedalgebra