VTRMC 2014

Evaluate $\int^2_0\frac{x(16-x^2)}{16-x^2+\sqrt{(4-x)(4+x)(12+x^2)}}dx$ .

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

Suppose we are given a $19\times19$ 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 $4\times1$ and $1\times4$ rectangles? (So that each of the $360$ squares is covered by exactly one rectangle.) Justify your answer.

Let $n\ge1$ and $r\ge2$ be positive integers. Prove that there is no integer $m$ such that $n(n+1)(n+2)=m^r$ .

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 $I\pmod3$ (so $\begin{pmatrix}a&bc&d\end{pmatrix}\in T$ means that $a,b,c,d\in\mathbb Z,ad-bc=1,$ and $3$ divides $b,c,a-1,d-1$ ).

(a) Let $f:T\to\mathbb R$ be a function such that for every $X,Y\in T$ with $Y\ne I$ , either $f(XY)>f(X)$ or $f(XY^{-1})>f(X)$ . 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 $b'=b$ . (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 $f(XY^{-1})>f(X)$ .

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 $B1=(10,3)$ . 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 $1\le i$ , $j\le2$ . In terms of inequalities, this means that $x_i<\min\{u_1,u_2\}$ and $y_i>\max\{z_1,z_2\}$ 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 $2\times2$ matrix $M=\begin{pmatrix}d(A_1,B_1)&d(A_1,B_2)d(A_2,B_1)&d(A_2,B_2)\end{pmatrix}$ . Prove that for any configuration of points $A_1,A_2,B_1,B_2$ as described before, $\det M>0$ .