Virginia Tech Regional Math Contest (2001)

Problem 1

Three infinitely long circular cylinders each with unit radius have their axes along the $x$, $y$ and $z$-axes. Determine the volume of the region common to all three cylinders. (Thus one needs the volume common to $\lbrace y^2 + z^2 \le 1\rbrace$, $\lbrace z^2 + x^2 \le 1\rbrace$, $\lbrace x^2 + y^2 \le 1\rbrace$.)

Problem 2

Two circles with radii 1 and 2 are placed so that they are tangent to each other and a straight line. A third circle is nestled between them so that it is tangent to the first two circles and the line. Find the radius of the third circle.

Problem 3

For each positive integer $n$, let $S_n$ denote the total number of squares in an $n \times n$ square grid. Thus $S_1 = 1$ and $S_2 = 5$, because a $2 \times 2$ square grid has four $1 \times 1$ squares and one $2 \times 2$ square. Find a recurrence relation for $S_n$, and use it to calculate the total number of squares on a chess board (i.e., determine $S_8$).

Problem 4

Let $a_n$ be the $n$th positive integer $k$ such that the greatest integer not exceeding $\sqrt{k}$ divides $k$, so the first few terms of ${a_n}$ are $\lbrace 1, 2, 3, 4, 6, 8, 9, 12, \dots \rbrace$. Find $a_{10000}$ and give reasons to substantiate your answer.

Problem 5

Determine the interval of convergence of the power series $\displaystyle{\sum_{n=1}^{\infty} \frac{n^n x^n}{n!}}$.

That is, determine the real numbers $x$ for which the above power series converges; you must determine correctly whether the series is convergent at the end points of the interval.

Problem 6

Find a function $f : \mathbb{R}^+ \to \mathbb{R}^+$ such that $f(f(x)) = \dfrac{3x + 1}{x + 3}$ for all positive real numbers $x$ (here $\mathbb{R}^+$ denotes the positive (nonzero) real numbers).

Problem 7

Let $G$ denote a set of invertible $2 \times 2$ matrices (matrices with complex numbers as entries and determinant nonzero) with the property that if $a, b$ are in $G$, then so are $ab$ and $a^{-1}$. Suppose there exists a function $f : G \to \mathbb{R}$ with the property that either $f(ga) > f(a)$ or $f(g^{-1} a) > f(a)$ for all $a, g$ in $G$ with $g \ne I$ (here $I$ denotes the identity matrix, $\mathbb{R}$ denotes the real numbers, and the inequality signs are strict inequality). Prove that given finite nonempty subsets $A, B$ of $G$, there is a matrix in $G$ which can be written in exactly one way in the form $xy$ with $x$ in $A$ and $y$ in $B$.