This post uses $\displaystyle \LaTeX$ and will be rendered with MathJax | The Virginia Tech Regional Mathematics Contest is sponsored each fall by the Mathematics Department at Virginia Tech. More than 130 colleges and universities throughout VA, DC, GA, IL, MD, NJ, NY, OH, NC, PA, SC, TN, WV and other states are invited to participate each year. Now approaching its 36th year, the contest began in 1979 and has grown to the point where over 100 schools with over 600 contestants participate in a typical year. Contestants at each participating school take the two and one-half hour exam on their own campus under the supervision of one of their own faculty members. Individuals compete for $\$750$ in regional prizes for which any contestant is eligible, and $\$250$ in local prizes for which only Virginia Tech students are eligible. This was given on Saturday, October 25, 2014 from 9:00 am to 11:30 am. |

**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, $\det M>0$.