Problem 1
Prove that $\sqrt{ab} \le (a + b)/2$ where $a$ and $b$ are positive real numbers.
Problem 2
Find the remainder $r$, $1 \le r \le 13$, when $2^{1985}$ is divided by $13$.
Problem 3
Find real numbers $c_1$ and $c_2$ so that
$$ I + c_1 M +c_2 M^2 = \begin{pmatrix} 0 & 0 \newline 0 & 0 \newline \end{pmatrix}, $$
where $M = \begin{pmatrix} 1 & 3 \newline 0 & 2 \newline \end{pmatrix}$ and $I$ is the identity matrix.
Problem 4
Consider an infinite sequence $\lbrace c_k \rbrace_{k=0}^\infty$ of circles. The largest, $C_0$, is centered at $(1, 1)$ and is tangent to both the $x$ and $y$-axes. Each smaller circle $C_n$ is centered on the line through $(1, 1)$ and $(2, 0)$ and is tangent to the next larger circle $C_{nā1}$ and to the $x$-axis. Denote the diameter of $C_n$ by $d_n$ for $n = 0, 1, 2, …$.
Find:
- $d_1$
- $\displaystyle{ \sum_{n=0}^\infty d_n }$
Problem 5
Find the function $f = f(x)$, defined and continuous on $\mathbb{R}^+ = \lbrace x \mid 0 \le x < \infty \rbrace$, that satisfies $f(x+1) = f(x) + x$ on $\mathbb{R}^+$ and $f(1) = 0$.
Problem 6
- Find an expression for $3/5$ as a finite sum of distinct reciprocals of positive integers. (For example: $2/7 = 1/7+1/8+1/56$.)
- Prove that any positive rational number can be so expressed.
Problem 7
Let $f = f(x)$ be a real function of a real variable which has continuous third derivative and which satisfies, for a given $c$ and all real $x$, $x \ne c$,
$$ \frac{f(x)ā f(c)}{x - c} = \frac{f’(x) + f’(c)}{2} . $$
Show that $\displaystyle{ f(x) = \frac{f’(x - f’(c))}{xāc} }$.
Problem 8
Let $p(x) = a_0 + a_1 x + \cdots + a_n x^n$, where the coefficients $a_i$ are real. Prove that $p(x) = 0$ has at least one root in the interval $0 \le x \le 1$ if $a_0 +a_1/2 + \cdots + a_n/(n+1) = 0$.