\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 2 Problems \\
  Year 16 --- Academic Year 2004--2005\\{\normalsize www.usamts.org}\end{center}}}
\begin{document}

\begin{description}
\item[1/2/16.]
\hspace{-2em}\parbox[t]{4.75in}{\hspace{2em}
The numbers 1 through 9 can be arranged in the triangles labeled
$a$ through $i$ illustrated on the right so that the numbers in each of the $2\times 2$ triangles sum to the same value $n$; that is
$${a+b+c+d} = {b+e+f+g} = {d+g+h+i} = n.$$
 For each possible sum $n$, show such an arrangement, labeled with the sum as shown at right. Prove that
there are no possible arrangements for any other values of  $n$. }\hspace{3pt}
\raisebox{-90pt}[0pt]{\setlength{\unitlength}{1pt}
\begin{picture}(90,90)
\put(0,15){\line(1,0){90}}
\put(15,40){\line(1,0){60}}
\put(30,65){\line(1,0){30}}
\put(0,15){\line(3,5){45}}
\put(30,15){\line(3,5){30}}
\put(60,15){\line(3,5){15}}
\put(90,15){\line(-3,5){45}}
\put(60,15){\line(-3,5){30}}
\put(30,15){\line(-3,5){15}}
\put(42,72){$a$}
\put(27,47){$b$}
\put(42,50){$c$}
\put(57,47){$d$}
\put(12,22){$e$}
\put(27,25){$f$}
\put(42,22){$g$}
\put(57,25){$h$}
\put(72,22){$i$}
\put(42,0){$n$}
\end{picture}}

\item[2/2/16.]
Call a number $a - b\sqrt{2}$ with $a$ and $b$ both positive integers \emph{tiny} if it is closer to zero than any  number $c - d\sqrt{2}$ such that $c$ and $d$ are positive integers  with $c<a$ and~$d<b$. Three numbers which are tiny are $1 - \sqrt{2}$,
$3 - 2\sqrt{2}$, and $7 - 5\sqrt{2}$. Without using a calculator or computer, prove 
whether or not each of the following is tiny:
$$(a)\ 58 - 41\sqrt 2\,, \hspace{1in}(b)\ 99 - 70\sqrt 2\,.$$

\item[3/2/16.]
A set is \emph{reciprocally whole} if its elements are distinct integers greater than 1 and the sum of the reciprocals of all those elements is exactly 1. Find a set $S$, as small as possible, that contains two reciprocally whole subsets, $I$ and $J$, which are distinct but not necessarily
disjoint (meaning they may share elements, but they may not be the same subset). Prove that no set with fewer elements
 than $S$ can contain two reciprocally whole subsets.

\item[4/2/16.]
\hspace{-2em}\parbox[t]{5.25in}{\hspace{2em}
How many quadrilaterals in the plane have four of the nine points $(0,0)$,
$(0,1)$, $(0,2)$, $(1,0)$, $(1,1)$, $(1,2)$, $(2,0)$, $(2,1$), $(2,2)$ as vertices? Do count both concave and convex quadrilaterals, but do not count figures where two sides cross each other or where a vertex angle
is $180^\circ$. Rigorously verify that no quadrilateral was skipped or
counted more than once.
\vfil}\hspace{10pt}
\raisebox{-35pt}[0pt]{\setlength{\unitlength}{1pt}
\begin{picture}(42,42)
\multiput(0,0)(19,0){3}{\circle*{2}}
\multiput(0,19)(19,0){3}{\circle*{2}}
\multiput(0,38)(19,0){3}{\circle*{2}}
\end{picture}}

\item[5/2/16.]
Two circles of equal radius can tightly fit inside right triangle $ABC$, which has $AB=13$, $BC=12$, and $CA=5$, in the three positions illustrated below. Determine the radii of the circles in each case.

\setlength{\unitlength}{1pt}
\begin{picture}(115,63)

%Note - this isn't quite a 5-12-13 triangle, but you don't have to
% have your triangle be exact!
% I would just build my diagrams in a graphing program rather than use TeX for 
% the diagrams in my solution.
\put(15,15){\line(1,0){90}}
\put(15,15){\line(5,2){90}}
\put(105,15){\line(0,1){36}}
\put(3,3){$B$}
\put(105,53){$A$}
\put(105,3){$C$}
\put(40,-10){Case (i)}

\put(72,26){\scalebox{0.915}{\begin{picture}(48,24)
\put(0,0){\circle{24}}\put(24,0){\circle{24}}
\end{picture}}}
\end{picture}
\hspace{0.5in}
\begin{picture}(115,63)
\put(15,15){\line(1,0){90}}
\put(15,15){\line(5,2){90}}
\put(105,15){\line(0,1){36}}
\put(3,3){$B$}
\put(105,53){$A$}
\put(105,3){$C$}
\put(40,-10){Case (ii)}
\put(97,23){\scalebox{0.8}{\begin{picture}(20,40)
\put(0,0){\circle{20}}\put(0,20){\circle{20}}
\end{picture}}}
\end{picture}
\hspace{0.5in}
\begin{picture}(115,63)
\put(15,15){\line(1,0){90}}
\put(15,15){\line(5,2){90}}
\put(105,15){\line(0,1){36}}
\put(3,3){$B$}
\put(105,53){$A$}
\put(105,3){$C$}
\put(40,-10){Case (iii)}
\put(73,26.2){\scalebox{0.93}{\begin{picture}(48,24)
\put(0,0){\circle{24}}\put(22.3,8.9){\circle{24}}
\end{picture}}}
\end{picture}

\end{description}

\noindent Round 2 Solutions must be submitted by {\bf November 22, 2004}. \\
\noindent Please visit 
{\bf http://www.usamts.org} for details about solution submission.
\\\noindent\copyright\ 2004 Art of Problem Solving Foundation

\pagebreak

Here are a couple examples to help you make diagrams with \LaTeX\ for Problem 4. 

\vskip0.2in


% Experiment by changing the numbers below and you will see how this works pretty quickly
% The next command will determine the size of your picture - change the 3 to make all following
% pictures larger or smaller.
% Students familiar with the xy package may choose to use that instead
\setlength{\unitlength}{3pt}

\begin{picture}(12,12) % (12,12) is size of picture

%start at coordinate (1,1), make 3 open  circles of radius 2, moving 5 to the right after each
\multiput(1,1)(5,0){3}{\circle{2}} 

%start at coordinate (1,6), make 3 open  circles of radius 2, moving 5 to the right after each
\multiput(1,6)(5,0){3}{\circle{2}}

%start at coordinate (1,11), make 3 open  circles of radius 2, moving 5 to the right after each
\multiput(1,11)(5,0){3}{\circle{2}}

%Make a filled circle of radius 2 at (1,1)
\put(1,1){\circle*{2}} 


%Make a filled circle of radius 2 at (11,1) 
\put(11,6){\circle*{2}}

%start at (1,1), make a line with slope 2/1 that goes 10x2 = 20 units right and 10x1 = 10 units up
\put(1,1){\line(2,1){10}}

\end{picture}

\vskip 0.5in


% Another example

\begin{picture}(12,12)

%build the open circles
\multiput(1,1)(5,0){3}{\circle{2}}
\multiput(1,6)(5,0){3}{\circle{2}}
\multiput(1,11)(5,0){3}{\circle{2}}

%build the closed circles (these will overwrite closed circles)
\put(6,6){\circle*{2}}
\put(11,6){\circle*{2}}
\put(6,11){\circle*{2}}

%diagonal line
\put(11,6){\line(-1,1){5}}

%horizontal line
\put(6,6){\line(1,0){5}}

%vertical line
\put(6,6){\line(0,1){5}}
\end{picture}

% Yes, this is a bit of a nutty system.  The xy package is a bit easier to use - if you know 
% how to install LaTeX packages, you may want to check that out.  You can also just build
% the pictures in a graphics program and include the image as described in the Pictures 
% section of the LaTeX pages on the website.

\end{document}
