VTRMC 2015

Find all n such that $n^{4}+6n^{3}+11n^{2}+3n+31$ is a perfect square.

Consider the harmonic series $\sum_{n\ge1}\frac1n=1+\frac12+\frac13+\ldots$ . 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 $\frac1k$ .)

Evaluate $\int^\infty_0\frac{\operatorname{arctan}(\pi x)-\operatorname{arctan}(x)}xdx$ (where $0\le\operatorname{arctan}(x)<\frac\pi2$ for $0\le x<\infty$ ).

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

1. $f(0,0)=1$ ;
2. $f(x,y)\ne0$ for only finitely many $(x,y)\in\mathbb R^2$ ;
3. $\sum_{r=1}^n\left|f(x+a_r,y+b_r)-f(x,y)\right|<\epsilon$ for every $(x,y)\in\mathbb R^2$ .

Justify your answer.

Let $n$ be a positive integer and let $x_1,\ldots,x_n$ 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 $\sum_{i=1}^na_ix_i$ 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 $D_1,D_2,\ldots$ of radius $R$ such that

(a) Each disk contains exactly two points of $S$ ; (b) Every point of $S$ lies in at least one disk; (c) Two distinct disks intersect in at most one point.