\def\frac#1#2{{{#1}\over{#2}}} \parindent=20 pt \parskip=10 pt \magnification=\magstep 2 {\sl These are the problems of the 1997 Texas A\& M University Freshman-Sophomore Contest. Your mission: to solve boldly where no solver has solved before. Please record both your answers, and the logical basis for your conclusions, on the data-storage sheets provided. Problems designated FS are for both freshmen and sophomores. Problems denoted F are for freshmen only; those denoted S are for sophomores only. All contestants compete in a single pool for prizes, though the problems set contestants at different stages are in part different. Calculators, pagers, etc. are not permitted. You have two hours.} \item{1FS.} A conical pile of sand grows beneath a sand-dumping hose. The hose drops a cubic foot of sand per second onto the pile. As the pile grows, sand shifts and slides down the sides so that it maintains its shape, with a slope of 45 degrees from horizontal. How many seconds must go by, from when the first sand spilled onto the flat dirt, before the sandpile height is growing at the rate of 1 inch per second? \item{2FS.} Let $f(x):=\int_0^x\sin^4t\, dt$. \itemitem{(a)} Find $f(2\pi)$. \itemitem{(b)} Sketch $f(x)$ over the interval $0\le x\le 2\pi$. \item{3FS.} Prove $$\eqalign{\sum_{n=1}^{\infty}&\frac{1}{n(n+1)}=1\cr \sum_{n=1}^{\infty}&\frac{1}{n(n+1)\dots (n+k)}=\frac{1}{k!k}\cr \sum_{n=1}^{\infty}&\frac{1}{n^2(n+1)\dots (n+k)}=\frac{\pi^2}{6}-\sum_{j=1}^k\frac{1}{j^2}\cr}$$ For the third identity, take as given that $\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$. \vfill\eject \item{4F.} A graduated cylinder in the chemistry storeroom has a leaky bottom. The cylinder was full of water yesterday evening at 5PM, but the next morning at 8AM, it was only half full. Water seeps out of the crack at the bottom at a rate proportional to the square root of the pressure. How many more hours, from 8 AM today, will it be until the cylinder is completely empty? Neglect evaporation, surface tension and so on. \item{5F.} Assume that $h(t)$ is positive for all $t\ge 0$, continuous, and has the property that for all $x>0$, $\int_x^{\infty}h(t)\, dt=2 h(x)$. Find $h(4)$. \item{6F.} A new delivery van costs $\$\,\!20000$. Once bought, there is a steady stream of maintenance costs. The company has a formula for the costs to be expected for a van aged $t$ years, over a short interval of further time $[t,t+\Delta t]$: $\$\,\!20t^2\Delta t$. Buying a new van every year, and scrapping the old one, is ridiculous. Holding on to a van that costs more in yearly maintenance than a new van is still sillier. The company wants to minimize the quantity: $${{\hbox{total cost of owning the van over\ } T \hbox{\ years}}\over{\hbox{number of years\ } T{\hbox{\ van is held}}}}$$At what age should an old van be scrapped and a new one bought?\vfill\eject \item{4S.} On the Mir space station, a pingpong ball is currently at $(0,0,0)$ and is moving forward along the $x$ axis. The paddle wielder wishes the ball to bounce off a paddle which is to be held motionless in some plane, so that the ball will bounce off and continue along the line $(x-2)/2=y/2=z$. Give the equation of the plane in which the paddle should be held. \item{5S.} Consider the set $S$ of all points in a cube which are closer to the center than to any of the corners. \itemitem{(a)} What fraction of the volume of the cube is in $S$? \itemitem{(b)} If now $T$ denotes the points of the cube which are closer to the center than to any of the {\it faces}, sketch the region $T$ and describe its boundary with an equation or set of equations. You have two choices for problem 6. Work one or the other, but not both. \item{6Sa.} Let $LB$ denote the both-directional Laplace transform: $LB$ applied to $y(t)$ gives the function $Y(s):=\int_{-\infty}^{\infty}\exp(-st)y(t)\, dt$. Take as given that $\int_{-\infty}^{\infty}\exp(-\pi t^2)\, dt=1$ (you may have seen this in connection with polar coordinates.) \itemitem{(a)} Find $LB(e^{-\pi t^2})$. \itemitem{(b)} Find $LB(f(t))$ if $f(t)=1-|t|$ for $|t|<1$ and $f(t)=0$ otherwise. \item{6Sb.} Consider the system of differential equations: $$\eqalign{y_1'&=y_2-y_1\cr y_2'&=y_3-y_2\cr y_3'&=y_1-y_3\cr}$$Your faithful assistant has turned the crank and got some responses from the machine; these are supplied in an abbreviated appendix (see next page). Assuming your assistant chose $m$ correctly, what was $m$? What were the eigenvalues of $m$? Discuss the behavior of the solutions as time passes: Is there a limit? Will the initially largest of the $y_k$'s always be the largest? \vfill\eject \bye Let $g(x):=\sum_{n=1}^{\infty}\frac{x^{n!}}{n!}$ whenever the series is convergent. \itemitem{(a)} Find integers $a$ and $b$ (explicitly) so that $$\left|\frac{a}{b}-f(1/2)\right|<10^{-10}$$ \itemitem{(b)} Find $f(1)$ exactly. \itemitem{(c)} Find the radius of convergence of the series.