## 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.

- Express $(I - A)^{-1}$ as a polynomial in $A$, where $I$ is the identity matrix.
- Find a $3 \times 3$ matrix satisfying $B^2 \ne 0$, $B^3 = 0$.

## 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 $G$.

- Show that $A B^2 = B^{-2} A$.
- Show that $A B^n = B^{-n} A$ for any integer $n$.
- Find $u$ and $v$ so that $(B^a A^b)(B^c A^d) = B^u A^v$.

## 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

- $a_1, a_2, a_3$ and $a_4$
- $a_{1981}$
- A simple general algorithm for evaluating $a_n$, for $n = 0, 1, …$

## Problem 8

Let

- $0 < a < 1$
- $0 < M_{k + 1} < M_k$, for $k = 0, 1, …$
- $\displaystyle{ \lim_{k \to \infty} M_k = 0 }$

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