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Functional Equations

Functional equations are equations where the unknowns are the functions. Rather than solving for x, you solve for the function in questions like "Find all functions that have these properties."

Level 4

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 \(f(-100)\).


This problem is posed by Zi Song Y.

\( f(x)\) 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 \( \displaystyle\lim_{n\rightarrow \infty }(\ln\: a_{n}) \).

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

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