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Extremes in functional equations

Solve the first problem 18 problems

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1. Iterated piecewise function value ●○○○○ 10 ptsFor integers $n$, let $f(n) = \begin{cases} n^2 + 1 & \text{if } n \text{ is odd}, \\ n^2 - 3n + 1 & \text{if } n \text{ is even}. \end{cases}$ Find $f(f(f(f(f(f(2))))))$.
2. Function value from a shifted definition ●○○○○ 10 ptsIf $f(x+1)=x$ for all real $x$, find $f(6)$.
3. Fixed point of a functional definition ●●○○○ 20 ptsLet $h(4x-1) = 2x + 7$ for all real $x$. For what value of $x$ is $h(x) = x$?
4. Maximum of a piecewise function ●●○○○ 20 ptsDefine a new operation $\oplus$ as follows: when $a \geq b$, $a \oplus b = a$; when $a < b$, $a \oplus b = b^2$. Let $f(x) = (1 \oplus x)\cdot x - (2 \oplus x)$ for $x \in [-2, 2]$. Find the maximum value of $f(x)$.
5. Integer point in a composed functional equation ●●●○○ 50 ptsThe function $f(x)$ satisfies $f(f(x)) = 6x - 2005$ for all real numbers $x$. There exists an integer $n$ such that $f(n) = 6n - 2005$. Find $n$.
6. Threshold of a functional equation ●●●○○ 50 ptsLet $f$ be a function taking the nonnegative integers to the positive integers such that $f(0) = f(1) = 1$ and $$\frac{f(n) f(m + 1)}{f(m)} + \frac{f(n) f(n - m)}{f(n - m - 1)} = f(n + 1)$$ for all nonnegative integers $n$ and $m$ with $n \ge m + 1$. Find the smallest nonnegative integer $n$ such that $f(n) > 10^6$.
7. Possible values of f(2018) ●●●○○ 50 ptsLet $f: \mathbb{Z} \rightarrow \mathbb{Z}$ be an even function with $f(1)=1$ and $f(2017) \neq 1$. For all integers $x, y$, $2f(x+y)-f(x)-f(y) \leqslant |f(x)-f(y)|$. Find the value of $f(2018)$.
8. Minimal f(2007) for a functional equation ●●●●○ 100 ptsLet $f$ be a function taking the positive integers to the positive integers, such that $f(mf(n)) = nf(m)$ for all positive integers $m$ and $n$. Find the smallest possible value of $f(2007)$.
9. Distinct values of a symmetric function ●●●●○ 100 ptsGiven a function $f$ for which $f(x) = f(398 - x) = f(2158 - x) = f(3214 - x)$ for all real $x$, what is the largest number of different values that can appear in the list $f(0), f(1), f(2), \ldots, f(999)$?
10. Smallest n with many solutions ●●●●○ 100 ptsLet $f(x)=|2\{x\}-1|$, where $\{x\}$ denotes the fractional part of $x$. Find the smallest positive integer $n$ such that the equation $n f(x f(x))=x$ has at least $2012$ real solutions.

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