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Problem 1

Show that the right circular cylinder of volume $V$ which has the least surface area is the one whose diameter is equal to its altitude. (The top and bottom are part of the surface.)

Problem 2

Let $S$ be a set which is closed under the binary operation $\circ$, with the following properties:

  1. there is an element $e \in S$ such that $a \circ e = e \circ a = a$, for each $a \in S$,

  2. $(a \circ b) \circ (c \circ d) = (a \circ c) \circ (b \circ d)$, for all $a, b, c, d \in S$.

Prove or disprove:

  1. $\circ$ is associative on $S$
  2. $\circ$ is commutative on $S$

Problem 3

Let $A$ be an $n \times n$ nonsingular matrix with complex elements, and let $\bar{A}$ be its complex conjugate. Let $B = A\bar{A} + I$, where $I$ is the $n \times n$ identity matrix.

Prove or disprove:

  1. $A^{-1}BA = \bar{B}$
  2. The determinant of $A \bar{A} + I$ is real.

Problem 4

Let $f(x)$ be continuously differentiable on $(0, \infty)$ and suppose $\displaystyle{ \lim_{x \to \infty} f’(x) = 0 }$.

Prove that $\displaystyle{ \lim_{x \to \infty} \frac{f(x)}{x} = 0 }$.

Problem 5

Show, for all positive integers $n = 1, 2,…$, that $14$ divides $3^{4n + 2} + 5^{2n + 1}$.

Problem 6

Suppose $a_n > 0$ and $\displaystyle{ \sum_{n = 1}^\infty a_n }$ diverges. Determine whether $\displaystyle{ \sum_{n = 1}^\infty \frac{a_n}{S_n^2} }$ converges, where $S_n = a_1 + a_2 + … + a_n$.

Problem 7

Let $S$ be a finite set of non-negative integers such that $| x - y| \in S$ for all $x, y \in S$.

  1. Give an example of such a set which contains ten elements.
  2. If $A$ is a subset of $S$ containing more than two-thirds of the elements of $S$, prove or disprove that every element of $S$ is the sum or difference of two elements from $A$.

Problem 8

Let $S$ be a finite set of polynomials in two variables, $x$ and $y$. For $n$ a positive integer, define $\Omega_n(S)$ to be the collection of all expressions $p_1 p_2 … p_k$, where $p_i \in S$ and $1 \le k \le n$. Let $d_n(S)$ indicate the maximum number of linearly independent polynomials in $\Omega_n(S)$. For example, $\Omega_2(\lbrace x^2, y \rbrace) = \lbrace x^2, y, x^2 y, x^4, y^2 \rbrace$ and $d_2(\lbrace x^2, y \rbrace) = 5$.

  1. Find $d_2(\lbrace 1, x, x + 1, y \rbrace)$.

  2. Find a closed formula in $n$ for $d_n(\lbrace 1, x, y \rbrace)$.

  3. Calculate the least upper bound over all such sets of $\displaystyle{ \limsup_{n \to \infty} \frac{\log d_n(S)}{\log n} }$.

    $\displaystyle{ \limsup_{n \to \infty} a_n = \lim_{n \to \infty}(\sup \lbrace a_n, a_{n + 1}, … \rbrace) }$, where $\sup$ means supremum or least upper bound.