Problem 1
It is known that $\displaystyle \int_1^2 \frac{\arctan(1+x)}{x} dx = q \pi \ln(2)$ for some rational number $q$. Determine $q$. Here $0 \le \arctan(x) < \pi/2$ for $0 \le x < \infty$.
Problem 2
Let $A, B \in M_6 (\mathbb{Z})$ such that $A \equiv I \equiv B \bmod 3$ and $A^3 B^3 A^3 = B^3$. Prove that $A = I$. Here $M_6 (\mathbb{Z})$ indicates the 6 by 6 matrices with integer entries, $I$ is the identity matrix, and $X \equiv Y \bmod 3$ means all entries of $X - Y$ are divisible by 3.
Problem 3
Prove that there is no function $f : \mathbb{N} \to \mathbb{N}$ such that $f (f (n)) = n + 1$. Here $\mathbb{N}$ is the positive integers ${1, 2, 3, \dots}$.
Problem 4
Let $m, n$ be integers such that $n \ge m \ge 1$. Prove that $\displaystyle \frac{\gcd(m, n)}{n} \binom{n}{m}$ is an integer. Here $\gcd$ denotes greatest common divisor and $\displaystyle \binom{n}{m} = \frac{n!}{m!(n-m)!}$ denotes the binomial coefficient.
Problem 5
For $n \in \mathbb{N}$, let $\displaystyle a_n = \int_0^{1/\sqrt{n}} |1 + e^{it} + e^{2it} + \dots + e^{nit}| dt$.
Determine whether the sequence $(a_n) = a_1, a_2, \dots$ is bounded.
Problem 6
For $n \in \mathbb{N}$, define
$$ \begin{align*} a_n &= \dfrac{1 + 1/3 + 1/5 + \dots + 1/(2n - 1)}{n+1} \text{ and } \newline b_n &= \dfrac{1/2 + 1/4 + 1/6 + \dots + 1/(2n)}{n}. \end{align*} $$
Find the maximum and minimum of $a_n - b_n$ for $1 \le n \le 999$.
Problem 7
A continuous function $f : [a, b] \to [a, b]$ is called piecewise monotone if $[a, b]$ can be subdivided into finitely many subintervals
$$ I_1 = [c_0, c_1], I_2 = [c_1, c_2], \dots, I_\ell = [c_{\ell-1}, c_\ell] $$
such that $f$ restricted to each interval $I_j$ is strictly monotone, either increasing or decreasing. Here we are assuming that $a = c_0 < c_1 < \dots < c_{\ell-1} < c_\ell = b$. We are also assuming that each $I_j$ is a maximal interval on which $f$ is strictly monotone. Such a maximal interval is called a lap of the function $f$, and the number $\ell = \ell(f)$ of distinct laps is called the lap number of $f$.
If $f : [a, b] \to [a, b]$ is a continuous piecewise-monotone function, show that the sequence $\left( \sqrt[n]{\ell(f^n)} \right)$ converges; here $f^n$ means $f$ composed with itself $n$-times, so $f^2(x) = f(f(x))$ etc.