Algebra
# Functional Equations

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

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

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

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

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

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

\[ f(f(x)) - x^{2} = xf(x).\]

Find \(f(-100)\).

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

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

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