## 36th Annual Virginia Tech Regional Mathematics Contest

This was given on Saturday, October 25, 2014 from 9:00 am to 11:30 am.

This post uses $\displaystyle \LaTeX$ and will be rendered using MathJax

This post uses $\displaystyle \LaTeX$ and will be rendered using MathJax

**1)**Find $\displaystyle \sum_{n=2}^{n=\infty }\frac{n^2-2n-4}{n^4+4n^2+16}$

**2)**Evaluate $\displaystyle \int_{0}^{2}\frac{\left ( 16-x^2 \right )x}{16-x^2+\sqrt{\left ( 4-x \right )\left ( 4+x \right )\left ( 12+x^2 \right )}}\: dx$

**3)**Find the least positive integer $n$ such that $2^{2014}$ divides $19^n-1$

**4)**Suppose we are given a 19 × 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 × 1 and 1 × 4 rectangles? (So each of the 360 squares is covered by exactly one rectangle.) Justify your answer.

**5)**

**Let $n\geq 1$ and $r\geq 2$ be positive integers. Prove that there is no integer $m$ such that $n\left ( n+1 \right )\left ( n+2 \right )=m^r$.**

**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 $I$ mod $3$ (so $\displaystyle \left ( \begin{matrix} a & b\\ c & d \end{matrix} \right )\in T$ means that $a,b,c,d\in \mathbb{Z}$, $ad-bc=1$, and $3$ divides $b,c,a-1,d-1$; "$\in$" means "is in").**

**(a)**

**Let $f:T\rightarrow \mathbb{R}$ (the real numbers) be a function such that for every $X,Y\in T$ with $Y\neq I$, either $f\left ( XY \right )>f\left ( X \right )$ or $f\left ( XY^{-1} \right )>f\left ( X \right )$ (or both). Show that given two finite nonempty subsets $A, B\; \text{of}\; T$, 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 $b'=b$.**

**(b)**Show that there is no $f:S\rightarrow \mathbb{R}$ such that for every $X,Y\in S$ with $Y\neq \pm I$, either $f\left ( XY \right )>f\left ( X \right )$ or $f\left ( XY^{-1} \right )>f\left ( X \right )$.

**7)**Let $A$, $B$ be two points in the plane with integer coordinates $\displaystyle A=\left ( x_1,y_1 \right )$ and $\displaystyle B=\left ( x_2,y_2 \right )$. (Thus $\displaystyle x_i,y_i\in\mathbb{Z}$, for $i=1,2$.) A path $\pi$: $A\rightarrow B$ 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 $\displaystyle A_1=\left ( 4,9 \right )$ to $\displaystyle B_1=\left ( 10,3 \right )$. The distance $d\left ( A,B \right )$ between $A$ and $B$ is the number of such paths. For example, the distance between $A=\left ( 0,2 \right )$ and $B=\left ( 2,0 \right )$ equals 6. Consider now two pairs of points in the plane $\displaystyle A_i=\left ( x_i,y_i \right )$ and $\displaystyle B_i=\left ( u_i,z_i \right )$ 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 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 $1\leq i$, $j\leq 2$. In terms of inequalities, this means that $x_i<\text{min}\left \{ u_1,u_2 \right \}$ and $y_i>\text{max}\left \{ z_1,z_2 \right \}$ for $i=1,2$.

**(a)**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 $B$.

**(b)**Consider the 2 × 2 matrix $\displaystyle M=\left ( \begin{matrix} d\left ( A_1,B_1 \right ) & d\left ( A_1,B_2 \right ) \\ d\left ( A_2,B_1 \right ) & d\left ( A_2,B_2 \right ) \end{matrix} \right )$. Prove that for any configuration of points $A_1,A_2,B_1,B_2$ as described before, $\det M>0$.