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\noindent PCMI-UFP \hfill Andrew J. Bernoff \\
Thursday, July 10, 2003 \hfill Harvey Mudd College\\

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 {\bf MAPLE Challenge Problems  } 
\end{center}

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These problems are designed to encourage the use of MAPLE, other computational aids, and web resources. 

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{\bf A1:} Consider the function
$$f_N(x) = \sum_{n=1}^N  \frac{\sin (nx)}{n}  \qquad  0 < x < \pi \quad .$$

a) Suppose these are partial sums of a Fourier series; can you guess the function $f(x)$ they are converging to? 

b) Determine $\lim_{N \to \infty} f_N(\pi/2) $. It may help if you evaluate  $\sin(nx)$ 
and let MAPLE evaluate the resulting infinite series.

c) Determine the value $x_*$ where $f_N(x)$ reaches a maximum. You should be able to find the answer explicitly.

d) Determine $\lim_{N \to \infty} f_N(x_*) $. You might try evaluating the limit as a Riemann sum.

e) Relate your result in (d) to Gibb's phenomena.

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{\bf A2:} The sequence ${a_n}$ satisfies the recurrence relationship
$$a_1=1 \qquad a_2 = 1 \qquad a_3=4 \qquad a_{n+3}  =2a_{n+2}+2a_{n+1} -a_{n}
 \quad {\rm for} \quad n=1,2,3, \ldots $$
Prove that $a_n$ is a perfect square for all $n$.

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{\bf A3:}  (a) What are the last two digits of  $3^{2003}$ ? $3^{2003^{2003}}$?

(b) What is the last non-zero digit of $2003!$ ~ ?


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{\bf A4:} Simplify the expression:
$$ x= \left ( 2 + \frac{10}{3\sqrt3} \right )^{1/3} + 
\left ( 2 - \frac{10}{3\sqrt3} \right )^{1/3}
$$
\phantom{shrdlu} \hfill (Hess)

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{\bf A5:} Find positive real numbers $x_1,x_2, \ldots x_n$  such that their sum is $2003$ and their product is as large as possible.

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{\bf A6:}  Let $x_1=1$ and for $m \ge 1$ let
$x_{m+1}= (m+3/2)^{-1} \sum_{k=1}^m x_k x_{m+1-k}$. Evaluate 
$\lim_{m\to \infty} x_m/x_{m+1}$. \\
\phantom{shrdlu} \hfill (American Mathematical  Monthly)

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{\bf A7:} What arrangement of the digits $0,1,2,3,4,5,6,7,8,9,0$ into the product 
of five two digit numbers yields the largest product? For example, 
$10 \times 23 \times 45 \times 67 \times 89 = 61,717,050.$\\
\phantom{shrdlu} \hfill (Kornhauser, Velleman \& Wagon)

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