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Problem 1

What is the remainder when $X^{1982} + 1$ is divided by $X - 1$? Verify your answer.

Problem 2

A box contains marbles, each of which is red, white or blue. The number of blue marbles is a least half the number of white marbles and at most one-third the number of red marbles. The number which are white or blue is at least $55$. Find the minimum possible number of red marbles.

Problem 3

Let $\bm{a}$, $\bm{b}$, and $\bm{c}$ be vectors such that $\lbrace \bm{a}, \bm{b}, \bm{c} \rbrace$ is linearly dependent. Show that

$$ \begin{vmatrix} \bm{a} \cdot \bm{a} & \bm{a} \cdot \bm{b} & \bm{a} \cdot \bm{c} \newline \bm{b} \cdot \bm{a} & \bm{b} \cdot \bm{b} & \bm{b} \cdot \bm{c} \newline \bm{c} \cdot \bm{a} & \bm{c} \cdot \bm{b} & \bm{c} \cdot \bm{c} \newline \end{vmatrix} = 0 $$

Problem 4

Prove that $t^{n - 1} + t^{1 - n} < t^n + t^{-n}$ when $t \ne 1$, $t > 0$ and $n$ is a positive integer.

Problem 5

When asked to state the Maclaurin Series, a student writes (incorrectly)

$$ (*) \quad f(x) = f(x) + x f’(x) + \frac{x^2}{2!} f’’(x) + \frac{x^3}{3!} f’’’(x) + … $$

  1. State Maclaurin’s Series for $f(x)$ correctly.
  2. Replace the left-hand side of $(*)$ by a simple closed form expression in $f$ in such a way that the statement becomes valid (in general).

Problem 6

Let $S$ be a set of positive integers and let $E$ be the operation on the set of subsets of $S$ defined by $EA = \lbrace x \in A \mid x \text{ is even} \rbrace$, where $A \subseteq S$. Let $CA$ denote the complement of $A$ in $S$. $ECEA$ will denote $E(C(EA))$, etc.

  1. Show that $ECECEA = EA$.
  2. Find the maximum number of distinct subsets of $S$ that can be generated by applying the operations $E$ and $C$ to a subset $A$ of $S$ an arbitrary number of times in any order.

Problem 7

Let $p(x) = ax^2 + bx + c$, where $a$, $b$ and $c$ are integers, with the property that $1 < p(1) < p(p(1)) < p(p(p(1)))$. Show that $a \ge 0$.

Problem 8

For $n \ge 2$, let $S_n = \displaystyle{ \sum_{k = n}^\infty \frac{1}{k^2} }$.

  1. Prove or disprove that $\dfrac{1}{n} < S_n < \dfrac{1}{n - 1}$.
  2. Prove or disprove that $S_n < \dfrac{1}{n - \frac{3}{4}}$.