Problem 1
Let $a, b$ be positive constants. Find the volume (in the first octant) which lies above the region in the $xy$-plane bounded by $x = 0$, $x = \pi/2$, $y = 0$, $y \sqrt{b^2 \cos^2 x + a^2 \sin^2 x} = 1$, and below the plane $z = y$.
Problem 2
Find rational numbers $a, b, c, d, e$ such that $$ \sqrt{7 + \sqrt{40}} = a + b\sqrt{2} + c\sqrt{5} + d\sqrt{7} + e\sqrt{10}. $$
Problem 3
Let $A$ and $B$ be nonempty subsets of $S = {1, 2, \dots, 99}$ (integers from 1 to 99 inclusive). Let $a$ and $b$ denote the number of elements in $A$ and $B$ respectively, and suppose $a + b = 100$. Prove that for each integer $s$ in $S$, there are integers $x$ in $A$ and $y$ in $B$ such that $x + y = s$ or $s + 99$.
Problem 4
Let $\lbrace 1, 2, 3, 4 \rbrace$ be a set of abstract symbols on which the associative binary operation $*$ is defined by the following operation table (associative means $(a * b) * c = a * (b * c)$):
$$ \begin{array}{|c|c|c|c|c|} \hline * & 1 & 2 & 3 & 4 \\ \hline 1 & 1 & 2 & 3 & 4 \\ \hline 2 & 2 & 1 & 4 & 3 \\ \hline 3 & 3 & 4 & 1 & 2 \\ \hline 4 & 4 & 3 & 2 & 1 \\ \hline \end{array} $$
If the operation $*$ is represented by juxtaposition, e.g., $2 * 3$ is written as $23$ etc., then it is easy to see from the table that of the four possible “words” of length two that can be formed using only 2 and 3, i.e., 22, 23, 32 and 33, exactly two, 22 and 33, are equal to 1. Find a formula for the number $A(n)$ of words of length $n$, formed by using only 2 and 3, that equal 1. From the table and the example just given for words of length two, it is clear that $A(1) = 0$ and $A(2) = 2$. Use the formula to find $A(12)$.
Problem 5
Let $n$ be a positive integer. A bit string of length $n$ is a sequence of $n$ numbers consisting of 0’s and 1’s. Let $f(n)$ denote the number of bit strings of length $n$ in which every 0 is surrounded by 1’s. (Thus for $n = 5$, 11101 is allowed, but 10011 and 10110 are not allowed, and we have $f(3) = 2$, $f(4) = 3$.) Prove that $f(n) < (1.7)^n$ for all $n$.
Problem 6
Let $S$ be a set of $2 \times 2$ matrices with complex numbers as entries, and let $T$ be the subset of $S$ consisting of matrices whose eigenvalues are $\pm 1$ (so the eigenvalues for each matrix in $T$ are $\lbrace 1, 1\rbrace$ or $\lbrace 1, -1\rbrace$ or $\lbrace -1, -1\rbrace$). Suppose there are exactly three matrices in $T$. Prove that there are matrices $A, B$ in $S$ such that $AB$ is not a matrix in $S$. (Note: $A = B$ is allowed).
Problem 7
Let $\lbrace a_n \rbrace_{n \ge 1}$ be an infinite sequence with $a_n \ge 0$ for all $n$. For $n \ge 1$, let $b_n$ denote the geometric mean of $a_1, \dots, a_n$, that is $(a_1 \dots a_n)^{1/n}$.
Suppose $\displaystyle{\sum_{n=1}^{\infty} a_n}$ is convergent. Prove that $\displaystyle{\sum_{n=1}^{\infty} b_n}$ is also convergent.