VTRMC 1979

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.)

Let $S$ be a set which is closed under the binary operation $\circ$ , with the following properties: (i) there is an element $e \in S$ such that $a \circ e = e \circ a = a$ , for each $a \in S$ . (ii) $(a \circ b) \circ (c \circ d)=(a \circ c) \circ (b \circ d)$ , for all $a,b, c,d \in S$ .

Prove or disprove: (a) $\circ$ is associative on S (b) $\circ$ is commutative on S

Let $A$ be an $n\times n$ nonsingular matrix with complex elements, and let $\overline{A}$ be its complex conjugate. Let $B = A\overline{A}+I$ , where $I$ is the $n\times n$ identity matrix. (a) Prove or disprove: $A^{-1}BA = \overline{B}$ . (b) Prove or disprove: the determinant of $A\overline{A}+I$ is real.

Let $f(x)$ be continuously differentiable on $(0,\infty)$ and suppose $\lim _ { x \rightarrow \infty } f ^ { \prime } ( x ) = 0$ . Prove that $\lim _ { x \rightarrow \infty } f ( x ) / x = 0$ .

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

Suppose $a _ { n } > 0$ and $\sum _ { n = 1 } ^ { \infty } a _ { n }$ diverges. Determine whether $\sum _ { n = 1 } ^ { \infty } a _ { n } / S _ { n } ^ { 2 }$ converges, where $S _ { n } = a _ { 1 } + a _ { 2 } + \dots + a _ { n } .$

Let S be a finite set of non-negative integers such that $| x - y | \in S$ whenever $x , y \in S$ . (a) Give an example of such a set which contains ten elements. (b) 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 $A$ .

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 $p _ { 1 } p _ { 2 } \dots p _ { k } ,$ where $p_i \in S$ and $1\leq k \leq n$ . Let $d_n(S)$ indicate the maximum number of linearly independent polynomials in $\Omega _ { n } ( S )$ . For example, $\Omega _ { 2 } \left( \left\{ x ^ { 2 } , y \right\} \right) = \left\{ x ^ { 2 } , y , x ^ { 2 } y , x ^ { 4 } , y ^ { 2 } \right\}$ and $d _ { 2 } \left( \left\{ x ^ { 2 } , y \right\} \right) = 5$ (a) Find $d _ { 2 } ( \{ 1 , x , x + 1 , y \} )$ . (b) Find a closed formula in $n$ for $d _ { n } ( \{ 1 , x , y \} )$ . (c) Calculate the least upper bound over all such sets of $\overline{\text{lim}} _ { n \rightarrow \infty } \frac { \log d _ { n } ( S ) } { \log n }$ ( $\overline{\text{lim}} _ { n \rightarrow \infty } a _ { n } = \lim _ { n \rightarrow \infty } ( \sup \left\{ a _ { n } , a _ { n + 1 } , \ldots \right\}$ , where sup means supremum or least upper bound.)