Virginia Tech Regional Math Contest (2015)

Problem 1

Find all integers $n$ for which $n^4 + 6n^3 + 11n^2 + 3n + 31$ is a perfect square.

Problem 2

The planar diagram below, with equilateral triangles and regular hexagons, sides length 2cm, is folded along the dashed edges of the polygons, to create a closed surface in three-dimensional Euclidean space. Edges on the periphery of the planar diagram are identified (or glued) with precisely one other edge on the periphery in a natural way. Thus for example, $BA$ will be joined to $QP$ and $AC$ will be joined to $DC$. Find the volume of the three-dimensional region enclosed by the resulting surface.

Problem 3

Let $(a_i)_{1 \le i \le 2015}$ be a sequence consisting of 2015 integers, and let $(k_i)_{1 \le i \le 2015}$ be a sequence of 2015 positive integers (positive integer excludes 0). Let

$$ A = \begin{pmatrix} a_{k_1}^1 & a_{k_1}^2 & \cdots & a_{k_1}^{2015} \newline a_{k_2}^1 & a_{k_2}^2 & \cdots & a_{k_2}^{2015} \newline \vdots & \vdots & \ddots & \vdots \newline a_{k_{2015}}^1 & a_{k_{2015}}^2 & \cdots & a_{k_{2015}}^{2015} \end{pmatrix} $$

Prove that $2015!$ divides $\det A$.

Problem 4

Consider the harmonic series $\displaystyle{\sum_{n\ge 1} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} \dots}$. Prove that every positive rational number can be obtained as an unordered partial sum of this series. (An unordered partial sum may skip some of the terms $\frac{1}{k}$.)

Problem 5

Evaluate $\displaystyle{\int_0^\infty \frac{\arctan(\pi x) - \arctan(x)}{x} dx}$ (where $0 \le \arctan(x) < \pi/2$ for $0 \le x < \infty$).

Problem 6

Let $(a_1, b_1), \dots, (a_n, b_n)$ be $n$ points in $\mathbb{R}^2$ (where $\mathbb{R}$ denotes the real numbers), and let $\varepsilon > 0$ be a positive number. Can we find a real-valued function $f(x, y)$ that satisfies the following three conditions?

Justify your answer.

Problem 7

Let $n$ be a positive integer and let $x_1, \dots, x_n$ be $n$ nonzero points in $\mathbb{R}^2$. Suppose $\langle x_i, x_j \rangle$ (scalar or dot product) is a rational number for all $i, j$ ($1 \le i, j \le n$). Let $S$ denote all points of $\mathbb{R}^2$ of the form $\sum_{i=1}^n a_i x_i$ where the $a_i$ are integers. A closed disk of radius $R$ and center $P$ is the set of points at distance at most $R$ from $P$ (includes the points distance $R$ from $P$). Prove that there exists a positive number $R$ and closed disks $D_1, D_2, \dots$ of radius $R$ such that