\documentclass[12pt]{article}
\usepackage{USAMTS_problems}
\chead{\parbox{1.6in}{\hskip-0.2in\scalebox{0.50}[0.50]{\includegraphics*[viewport=120 340 700 500]{USAMTSBW.pdf}}}\parbox{5in}{\begin{center}\large {\bf USA Mathematical Talent Search}
\\ Round 3 Problems \\
Year 16 --- Academic Year 2004--2005\\{\normalsize www.usamts.org}\end{center}}}
\begin{document}
Please follow the rules below to ensure that your paper is graded properly.
\begin{enumerate}
\item \label{imp}Put your name, username, and USAMTS ID$\#$ on {\bf every page you submit}.
\item Once you send in your solutions, that submission is final. You cannot resubmit solutions.
\item If you have already sent in an Entry Form and a Permission Form, you do not need to resend them.
\item Confirm that your email address in your USAMTS Profile is
correct. You can do so by logging into the site, then clicking
on My USAMTS on the sidebar, then click Profile. If you are registered for the USAMTS and haven't received
any email from us about the USAMTS, your email address is probably wrong in your Profile.
\item Do not fax solutions written in pencil.
\item No single page should contain solutions to more than one problem.
\item By the end of December, Round 2 results will be posted at www.usamts.org. To see your
results, log in to the USAMTS page, then go to My USAMTS.
\item Submit your solutions by January 3, 2005 (postmark deadline), via one of the methods below.
\begin{enumerate}
\item Email: solutions@usamts.org. Please see usamts.org for a list of acceptable file types.
Do not send .doc Microsoft Word files.
\item Fax: (619) 445-2379
\item Snail mail: USAMTS, P.O. Box 2090, Alpine, CA 91903--2090.
\end{enumerate}
\item Re--read item~\ref{imp}.
\end{enumerate}
\pagebreak
\USprob{1/3/16.}{
\rightskip 0in
Given two integers $x$ and $y$, let $(x \| y)$ denote the \emph{concatenation} of $x$ by $y$, which is obtained by appending the digits of $y$ onto the end of $x$. For example, if $x=218$ and $y=392$, then $(x \| y) = 218392$. \\
(a) Find 3-digit integers $x$ and $y$ such that $6(x \| y) = (y \| x)$. \\
(b) Find 9-digit integers $x$ and $y$ such that $6(x \| y) = (y \| x)$.
}
\vskip0.25in
\USprob{2/3/16.}{
\rightskip 0in
Find three isosceles triangles, no two of which are congruent, with integer sides, such that each triangle's area is numerically equal to $6$ times its perimeter.
}
\vskip0.25in
\USprob{3/3/16.}{
\rightskip 0in
Define the recursive sequence 1, 4, 13, $\ldots$ by $s_1 = 1$ and $s_{n+1} = 3s_n + 1$ for all positive integers $n$. The element $s_{18} = 193710244$ ends in two identical digits. Prove that all the elements in the sequence that end in two or more identical digits come in groups of three consecutive elements that have the same number of identical digits at the end.
}
\vskip0.25in
\USprob{4/3/16.}{
\rightskip 1.5in
Region $ABCDEFGHIJ$ consists of 13 equal squares and is inscribed in rectangle $PQRS$ with $A$ on $\overline{PQ}$, $B$ on $\overline{QR}$, $E$ on $\overline{RS}$, and $H$ on $\overline{SP}$, as shown in the figure on the right. Given that $PQ=28$ and $QR=26$, determine, with proof, the area of region $ABCDEFGHIJ$.}
\vskip-0.75in\hskip5in
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\begin{picture}(84, 90)
\put(28,13){\line(4,-1){12}}
\put(31,25){\line(4,-1){12}}
\put(10,43){\line(4,-1){60}}
\put(13,55){\line(4,-1){60}}
\put(16,67){\line(4,-1){36}}
\put(19,79){\line(4,-1){36}}
\put(10,43){\line(1,4){9}}
\put(22,40){\line(1,4){9}}
\put(28,13){\line(1,4){15}}
\put(40,10){\line(1,4){15}}
\put(58,31){\line(1,4){3}}
\put(70,28){\line(1,4){3}}
\put(10,10){\framebox(63,69){}}
\put(0,0){$P$}
\put(0,81){$Q$}
\put(73,81){$R$}
\put(73,0){$S$}
\put(0,40){$A$}
\put(15,81){$B$}
\put(56,69){$C$}
\put(50,47){$D$}
\put(73,37){$E$}
\put(63,18){$F$}
\put(45,22){$G$}
\put(33,0){$H$}
\put(21,11){$I$}
\put(25,27){$J$}
\end{picture}
\USprob{5/3/16.}{
\rightskip 2in
Consider an isosceles triangle $ABC$ with side lengths $AB = AC = 10\sqrt{2}$ and $BC =10\sqrt{3}$. Construct semicircles $P$, $Q$, and $R$ with diameters $AB$, $AC$, $BC$ respectively, such that the plane of each semicircle is perpendicular to the plane of $ABC$, and all semicircles are on the same side of plane $ABC$ as shown. There exists a plane above triangle $ABC$ that is tangent to all three semicircles $P$, $Q$, $R$ at the points $D$, $E$, and $F$ respectively, as shown in the diagram. Calculate, with proof, the area of triangle $DEF$.
\vskip-2in\hskip4.75in
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\includegraphics[viewport=2.5in 4in 6.5in 7.5in,scale=0.5,clip]{3Ddiagram.pdf}
\end{picture}
}
\rightskip0in
\noindent Round 3 Solutions must be submitted by {\bf January 3, 2005}. \\
\noindent Please visit
{\bf http://www.usamts.org} for details about solution submission.
\\\noindent\copyright\ 2004 Art of Problem Solving Foundation
\end{document}