Let $f$ be a function from $\mathbb{N}$ to $\mathbb{N}$ such that

(i) $f(1)=0$

(ii) $f(2n)=2f(n)+1)$

(iii) $f(2x+1)=2f(n)$

Find the least value of $n$ such that $f(n)=2016$.

Find the number of functions $f(x)$ from $\{1, 2, 3, 4, 5\}$ to $\{1, 2, 3, 4, 5\}$ that satisfy $f(f(x)) = f(f(f(x)))$ for all $x$ in $\{1, 2, 3, 4, 5\}$.

Find the number of functions $f$ from $\{0, 1, 2, 3, 4, 5, 6\}$ to the integers such that $f(0) = 0$, $f(6) = 12$, and $|x - y|$ $\leq$ $|f(x) - f(y)|$ $\leq$ $3|x - y|$ for all $x$ and $y$ in $\{0, 1, 2, 3, 4, 5, 6\}$.

Construct one polynomial $f(x)$ with real coefficients and with all of the following properties:

• it is an even function
• $f(2)=f(-2)=0$
• $f(x) > 0$ when $-2 < x < 2$, and
• the maximum of $f(x)$ is achieved at $x=\pm 1$.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a periodic continuous function of period $T > 0$, that is $f(x+T)=f(x)$ holds for any $x\in\mathbb{R}$. Show that

$$\lim_{x\to\infty}\frac{1}{x}\int_0^xf(t)dt=\frac{1}{T}\int_0^Tf(t)dt$$

Let $f_n (x) = (2 + (−2)^n ) x^2 + (n + 3) x + n^2$.

1. Write down $f_3(x)$ and find its maximum value. Also determine for what value of $n$ does the function $f_n(x)$ have a maximum value (as $x$ varies). You do not need to compute this maximum value.
2. Write down $f_1(x)$. Calculate $f_1(f_1(x))$ and $f_1(f_1(f_1(x)))$. Find an expression, simplified as much as possible, for $$\underbrace{f_1(f_1(\cdots f_1(x)))}_{k}$$
3. Write down $f_2(x)$. Find the degree of the function $$\underbrace{f_2(f_2(\cdots f_2(x)))}_{k}$$

Let $f(x)$ be a second degree function satisfying $f(-2)=0$ and $2x \lt f(x) \le\frac{x^2+4}{2}$. Find the value of $f(10)$.