Problem 1
Find the inflection point of the graph of $F(x) = \displaystyle{\int_{0}^{x^3} e^{t^2} dt}$, for $x \in \mathbb{R}$.
Problem 2
Assume that $x_1 > y_1 > 0$ and $y_2 > x_2 > 0$. Find a formula for the shortest length $l$ of a planar path that goes from $(x_1, y_1)$ to $(x_2, y_2)$ and that touches both the $x$-axis and the $y$-axis. Justify your answer.
Problem 3
Let $f_n(x)$ be defined recursively by $$ f_0(x) = x, \quad f_1(x) = f(x), \quad f_{n+1}(x) = f(f_n(x)), \quad \text{for } n \ge 0, $$ where $f(x) = 1 + \sin(x - 1)$.
- Show that there is a unique point $x_0$ such that $f_2(x_0) = x_0$.
- Find $\displaystyle{\sum_{n=0}^{\infty} \frac{f_n(x_0)}{3^n}}$ with the above $x_0$.
Problem 4
Let $\lbrace t_n \rbrace_{n=1}^{\infty}$ be a sequence of positive numbers such that $t_1 = 1$ and $t_{n+1}^2 = 1 + t_n$, for $n \ge 1$. Show that $t_n$ is increasing in $n$ and find $\displaystyle{\lim_{n \to \infty} t_n}$.
Problem 5
Let $A = \begin{pmatrix} 0 & -2 \newline 1 & 3 \end{pmatrix}$. Find $A^{100}$. You have to find all four entries.
Problem 6
Let $p(x)$ be the polynomial $p(x) = x^3 + ax^2 + bx + c$. Show that if $p(r) = 0$, then $$ \frac{p(x)}{x-r} - 2\frac{p(x+1)}{x+1-r} + \frac{p(x+2)}{x+2-r} = 2 $$ for all $x$ except $x = r$, $r - 1$ and $r - 2$.
Problem 7
Find $\displaystyle{\lim_{n \to \infty} \frac{2 \log 2 + 3 \log 3 + \dots + n \log n}{n^2 \log n}}$.
Problem 8
Some goblins, $N$ in number, are standing in a row while “trick-or-treat”-ing. Each goblin is at all times either $2'$ tall or $3'$ tall, but can change spontaneously from one of these two heights to the other at will. While lined up in such a row, a goblin is called a Local Giant Goblin (LGG) if he/she/it is not standing beside a taller goblin. Let $G(N)$ be the total of all occurrences of LGG’s as the row of $N$ goblins transmogrifies through all possible distinct configurations, where height is the only distinguishing characteristic. As an example, with $N = 2$, the distinct configurations are $\hat{2}\hat{2}$, $2\hat{3}$, $\hat{3}2$, $\hat{3}\hat{3}$, where a cap indicates an LGG. Thus $G(2) = 6$.
- Find $G(3)$ and $G(4)$.
- Find, with proof, the general formula for $G(N)$, $N = 1, 2, 3, \dots$.