Functional Equations

Functional Equations: Level 4 Challenges


Let F(x) F(x) be a non-decreasing function defined on [0,1] [0, 1 ] such that

{2F(x3)=F(x),F(x)+F(1x)=1. \begin{cases} 2 F \left ( \frac{x}{3} \right) = F(x), \\ F(x) + F(1-x) = 1 .\\ \end{cases}

What is the value of F(113) F ( \frac{1}{13} )?

Find the number of functions f:RR f : \mathbb{R} \rightarrow \mathbb{R} such that f(x+y)=f(x)f(y)f(xy)f(x+y)=f(x)\cdot f(y)\cdot f(xy) for all x,yx,y in R \mathbb{R} .

Note: R \mathbb{R} denotes the set of real numbers.

Let f(x)f(x) be a polynomial such that

f(f(x))x2=xf(x). f(f(x)) - x^{2} = xf(x).

Find f(100)f(-100).

This problem is posed by Zi Song Y.

f(x) f(x) is an even function with codomain R.\mathbb{R}. Its graph is symmetric about the line x=1 x=1 , and f(x1+x2)=f(x1)f(x2)f(x_{1}+x_{2})=f(x_{1})\cdot f(x_{2}) for any x1,x2[0,12] x_{1},x_{2}\in [0,\frac{1}{2}] , and f(1)>0. f(1)> 0.

Let an=f(2n+12n) a_{n}=f(2n+\frac{1}{2n}) . Find the value of limn(lnan) \displaystyle\lim_{n\rightarrow \infty }(\ln\: a_{n}) .

Find the number of functions such that f:RRf:\mathbb{R}\rightarrow\mathbb{R} satisfying f(x2+yf(z))=xf(x)+zf(y)f(x^{2} + yf(z))=xf(x) + zf(y) for all x,y,zRx,y,z\in \mathbb{R}.


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