VTRMC 1980

Let $*$ denote a binary operation on a set $S$ with the property that $$(w*x)*(y*z) = w * z$$ for all $w,x,y,z\in S.$ Show

(a) If $a*b=c,$ then $c*c = c.$ (b) If $a*b=c,$ then $a*x=c*x$ for all $x\in S.$

The sum of the first $n$ terms of the sequence $$1,1+2,1+2+2^2,\ldots,1+2+\cdots+2^{k-1},\ldots$$ is of the form $2^{n+R}+Sn^2+Tn+U$ for all $n>0.$ Find $R,S,T,$ and $U.$

Let $$a_n = \frac{1\cdot3\cdot5\cdot\cdots\cdot(2n-1)}{2\cdot4\cdot6\cdot\cdots\cdot2n}.$$ (a) Prove that $\lim_{n\to \infty}a_n$ exists. (b) Show that $$a_n = \frac{\left(1-\frac1{2^2}\right)\left(1-\frac1{4^2}\right)\left(1-\frac1{6^2}\right)\cdots\left(1-\frac{1}{(2n)^2}\right)}{(2n+1)a_n}.$$ (c) Find $\lim_{n\to\infty}a_n$ and justify your answer

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 $\int_{-1}^1 P(x) \ dx = P(x_1) + P(x_2),$ where $x_1$ and $x_2$ are independent of the polynomial $P.$ (a) Show that $x_1=-x_2.$ (b) Find $x_1$ and $x_2.$

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

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

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

(a) Determine whether and arbitrary $O\in O'$ is finite. (b) Determine whether the carnality $|O|$ of $O$ is bounded for $O\in O'.$ (c) Determine whether $|O|$ can be countable infinite for any $O\in O'.$

Let $z=x+iy$ be a complex number with $x$ and $y$ rational and with $|z| = 1.$ (a) Find two such complex numbers. (b) Show that $|z^{2n}-1|=2|\sin n\theta|,$ where $z=e^{i\theta}.$ (c) Show that $|z^2n -1|$ is rational for every $n.$