Problem 1
Find $\displaystyle \sum_{n=2}^\infty \frac{n^2 - 2n - 4}{n^4 + 4n^2 + 16}$.
Problem 2
Evaluate $\displaystyle \int_0^2 \frac{(16-x^2)x}{16-x^2 + \sqrt{(4-x)(4+x)(12+x^2)}} dx$.
Problem 3
Find the least positive integer $n$ such that $2^{2014}$ divides $19^n - 1$.
Problem 4
Suppose we are given a $19 \times 19$ chessboard (a table with $19^2$ squares) and remove the central square. Is it possible to tile the remaining $19^2 - 1 = 360$ squares with $4 \times 1$ and $1 \times 4$ rectangles? (So each of the 360 squares is covered by exactly one rectangle.) Justify your answer.
Problem 5
Let $n \ge 1$ and $r \ge 2$ be positive integers. Prove that there is no integer $m$ such that $n(n + 1)(n + 2) = m^r$.
Problem 6
Let $S$ denote the set of 2 by 2 matrices with integer entries and determinant 1, and let $T$ denote those matrices of $S$ which are congruent to the identity matrix $I \bmod 3$ (so $\begin{pmatrix} a & b \ c & d \end{pmatrix} \in T$ means that $a, b, c, d \in \mathbb{Z}$, $ad - bc = 1$, and 3 divides $b, c, a - 1, d - 1$).
Let $f : T \to \mathbb{R}$ (the real numbers) be a function such that for every $X, Y \in T$ with $Y \ne I$, either $f (XY ) > f (X)$ or $f (XY^{-1}) > f (X)$ (or both). Show that given two finite nonempty subsets $A, B$ of $T$, there are matrices $a \in A$ and $b \in B$ such that if $a' \in A$, $b' \in B$ and $a' b' = ab$, then $a' = a$ and $b' = b$.
Show that there is no $f : S \to \mathbb{R}$ such that for every $X, Y \in S$ with $Y \ne \pm I$, either $f (XY ) > f (X)$ or $f (XY^{-1}) > f (X)$.
Problem 7
Let $A, B$ be two points in the plane with integer coordinates $A = (x_1, y_1)$ and $B = (x_2, y_2)$. (Thus $x_i, y_i \in \mathbb{Z}$, for $i = 1, 2$.) A path $\pi : A \to B$ is a sequence of down and right steps, where each step has an integer length, and the initial step starts from $A$, the last step ending at $B$. In the figure below, we indicated a path from $A_1 = (4, 9)$ to $B_1 = (10, 3)$. The distance $d(A, B)$ between $A$ and $B$ is the number of such paths. For example, the distance between $A = (0, 2)$ and $B = (2, 0)$ equals 6. Consider now two pairs of points in the plane $A_i = (x_i, y_i)$ and $B_i = (u_i, z_i)$ for $i = 1, 2$, with integer coordinates, and in the configuration shown in the picture (but with arbitrary coordinates):
- $x_2 < x_1$ and $y_1 > y_2$, which means that $A_1$ is North-East of $A_2$; $u_2 < u_1$ and $z_1 > z_2$, which means that $B_1$ is North-East of $B_2$.
- Each of the points $A_i$ is North-West of the points $B_j$, for $1 \le i, j \le 2$. In terms of inequalities, this means that $x_i < \min\lbrace u_1, u_2 \rbrace$ and $y_i > \max\lbrace z_1, z_2 \rbrace$ for $i = 1, 2$.
Find the distance between two points $A$ and $B$ as before, as a function of the coordinates of $A$ and $B$. Assume that $A$ is North-West of $B$.
Consider the $2 \times 2$ matrix $M = \begin{pmatrix} d(A_1, B_1) & d(A_1, B_2) \newline d(A_2, B_1) & d(A_2, B_2) \end{pmatrix}$. Prove that for any configuration of points $A_1, A_2, B_1, B_2$ as described before, $\det M > 0$.