\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 1 Problems \\
  Year 16 --- Academic Year 2004--2005\\{\normalsize www.usamts.org}\end{center}}}
\begin{document}


\parbox{6in}{

\USprob{1/1/16.}{ 
\rightskip 2in
The numbers $1$ through $10$ can be arranged along the vertices and sides of a pentagon so that the sum of the three numbers along each side is the same. The diagram on the right shows an arrangement with sum~$16$. Find, with proof, the smallest possible value for a sum and give an example of an arrangement with that sum.
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\begin{picture}(90,80)
\put(15,46){\line(1,-3){12}}
\put(27,10){\line(1,0){38}}
\put(65,10){\line(1,3){12}}
\put(15,46){\line(4,3){31}}
\put(77,46){\line(-4,3){31}}
\put(20,0){4}
\put(43,0){9}
\put(65,0){3}
\put(73,22){6}
\put(80,44){7}
\put(63,59){8}
\put(43,71){1}
\put(22,59){5}
\put(1,44){10}
\put(12,22){2}
\end{picture}}}

\vskip0.25in
\USprob{2/1/16.} {
\rightskip 0in
For the equation
\begin{displaymath}
(3x^2 + y^2 - 4y - 17)^3 - (2x^2 + 2y^2 - 4y - 6)^3 = (x^2 - y^2 - 11)^3,
\end{displaymath}
determine its solutions $(x,y)$ where both $x$ and $y$ are integers. Prove that your answer lists all the integer solutions.}

\vskip0.25in\USprob{3/1/16.}{Given that $5r + 4s + 3t + 6u = 100$, where $r \geq s \geq t \geq u \geq 0$ are real numbers, find, with proof, the maximum and minimum possible values of $r + s + t + u$.}

\vskip0.25in\USprob{4/1/16.}{The interior angles of a convex polygon form an arithmetic progression with a common difference of $4^\circ$. Determine the number of sides of the polygon if its largest interior angle is~$172^\circ$.}

\vskip0.25in\USprob{5/1/16.}{\rightskip 1.7in 
Point~$G$ is where the medians of the triangle $ABC$ intersect and point~$D$ is the midpoint of side~$\overline{BC}$.
The triangle $BDG$ is equilateral with side length~l. Determine the lengths, $AB$, $BC$, and $CA$, of the sides of triangle~$ABC$.}
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\put(15,0){\line(1,0){60}}
\put(15,0){\line(-1,3){15}}
\put(15,0){\line(1,1){22.5}}
\put(45,0){\line(-1,1){45}}
\put(75,0){\line(-5,3){75}}
\put(75,0){\line(-3,1){67.5}}
\end{picture}}}
\put(0,87){$A$}
\put(5,48){$F$}
\put(15,0){$B$}
\put(50,0){$D$}
\put(85,0){$C$}
\put(49,50){$E$}
\put(43,36){$G$}
\put(37,4){\scriptsize 1}
\put(27,22){\scriptsize 1}
\put(49,21){\scriptsize 1}
\end{picture}}


}

\vfill
\noindent\hrulefill 

\noindent Round 1 Solutions must be submitted by {\bf October 4, 2004}. \\
\noindent Please visit 
{\bf http://www.usamts.org} for details about solution submission.
\\\noindent\copyright\ 2004 Art of Problem Solving Foundation

\end{document}