Virginia Tech Regional Math Contest (1980)

Problem 1

Let $\ast$ denote a binary operation on a set $S$ with the property that

$$(w \ast x) \ast (y \ast z) = w \ast z \text{ for all } w, x, y, z \in S$$

Show:

Problem 2

The sum of the first $n$ terms of the sequence

$$1, (1 + 2), (1 + 2 + 2^2), …, (1 + 2 + … + 2^{k - 1}), …$$

is of the form $2^{n + R} + Sn^2 + Tn + U$ for all $n > 0$. Find $R$, $S$, $T$ and $U$.

Problem 3

Let $\displaystyle{ a_n = \frac{1 \cdot 3 \cdot 5 \cdot \ … \ \cdot (2n - 1)} {2 \cdot 4 \cdot 6 \cdot \ … \ \cdot 2n} }$.

Problem 4

Let $P(x)$ be any polynomial of degree at most $3$. It can be shown that there are numbers $x_1$ and $x_2$ such that $\displaystyle{\int_{-1}^{1} P(x) dx = P(x_1) + P(x_2)}$, where $x_1$ and $x_2$ are independent of the polynomial $P$.

Problem 5

For $x > 0$, show that $e^x < (1 + x)^{1 + x}$.

Problem 6

Given the linear fractional transformation of $x$ into $\displaystyle{ f_1(x) = \frac{2x - 1}{x + 1} }$, define $f_{n + 1}(x) = f_1(f_n(x))$ for $n = 1, 2, 3, …$. It can be shown that $f_{35} = f_5$. Determine $A$, $B$, $C$, and $D$ so that $\displaystyle{ f_{28}(x) = \frac{Ax + B}{Cx + D} }$.

Problem 7

Let $S$ be the set of all ordered pairs of integers $(m, n)$ satisfying $m > 0$ and $n < 0$. Let $\langle$ be a partial ordering on $S$ defined by the statement: $(m, n) \ \langle\ (m', n')$ if and only if $m \le m'$ and $n \le n'$. An example is $(5, - 10) \ \langle\ (8, - 2)$. Now let $O$ be a completely ordered subset of $S$, i.e., if $(a, b) \in O$ and $(c, d ) \in O$, then $(a, b) \ \langle\ (c, d )$ or $(c, d ) \ \langle\ (a, b)$. Also let $C$ denote the collection of all such completely ordered sets.

Problem 8

Let $z = x + iy$ be a complex number with $x$ and $y$ rational and with $|z| = 1$.