Virginia Tech Regional Math Contest (1981)

Problem 1

The number $2^{48} - 1$ is exactly divisible by what two numbers between $60$ and $70$?

Problem 2

For which real numbers $b$ does the function $f(x)$, defined by the conditions $f(0) = b$ and $f' = 2f - x$, satisfy $f(x) > 0$ for all $x \ge 0$?

Problem 3

Let $A$ be non-zero square matrix with the property that $A^3 = 0$, where $0$ is the zero matrix, but with $A$ being otherwise arbitrary.

Problem 4

Let $\displaystyle{P(x) = \sum_{n=0}^\infty F_n x^n}$ (wherever the series converges), where $F_n$ is the $n$th Fibonacci number defined by $F_0 = F_1 = 1, F_n = F_{n - 1} + F_{n - 2}, n > 1$. Find an explicit closed form for $P(x)$.

Problem 5

Two elements $A$, $B$ in a group $G$ have the property $ABA^{-1}B = 1$, where $1$ denotes the identity element in $G$.

Problem 6

With $k$ a positive integer, prove that $\left( 1 - \dfrac{1}{k^2} \right)^k \ge 1 - \dfrac{1}{k}$.

Problem 7

Let $A = \lbrace a_0, a_1, … \rbrace$ be a sequence of real numbers and define the sequence $A' = {a_0', a_1', …}$ as follows for $n = 0, 1, …$: $a_{2n}' = a_n$, $a_{2n + 1}' = a_n + 1$. If $a_0 = 1$ and $A' = A$, find

Problem 8

Let

If $\displaystyle{ b_n = \sum_{k = 0}^\infty a^{n - k} M_k }$, prove that $\displaystyle{ \lim_{n \to \infty} b_n = 0 }$.