Algebra
# Functional Equations

Find all functions \( f: \mathbb{R} \backslash \{ 0, 1 \} \rightarrow \mathbb{R} \) such that

\[ f \left( \frac{a}{a-1} \right) - 11 f \left( \frac{ a-1} {a} \right) = 0, \]

for \( a \neq 0, 1 \).

A function \(f(x)\) from the positive integers to the reals satisfies

1) \( f(1) = 792 \)

2) \( f(1) + f(2) + \ldots + f(n) = n^2 f(n) \) for any positive integer \(n\).

What is the value of \( f(11) \)?

Consider the function \(f(x)\) satisfying

\[\left(x^2-x+1\right)f\left(x^2\right)=f(x), f(3)=3.\]

If the value of \(f(-9)\) can be expressed as \(\frac{a}{b},\) where \(a\) and \(b\) are coprime positive integers, what is the value of \(a+7b?\)

A function \( f: \mathbb{R} \backslash \{ 0, 1 \} \rightarrow \mathbb{R} \) satisfies

\[ f(x) + f \left( \frac{1}{1-x} \right) = 5 x,\]

for \( x \neq 0, 1 \). What is the value of \( 40 f( 5)\)?