\documentclass[12pt]{article} \usepackage{amsfonts} \setlength{\textheight}{9.25in} \setlength{\textwidth}{6.5in} \setlength{\topmargin}{0.0in} \setlength{\headheight}{0.0in} \setlength{\headsep}{0.0in} \setlength{\leftmargin}{0.0in} \setlength{\oddsidemargin}{0.0in} \setlength{\parindent}{1pc} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calO{\mathcal{O}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\ida{\mathfrak{a}} \def\idb{\mathfrak{b}} \def\idm{\mathfrak{m}} \def\idn{\mathfrak{n}} \def\idp{\mathfrak{p}} \def\CC{\mathbb{C}} \def\FF{\mathbb{F}} \def\N{\mathbb{N}} \def\NN{\mathbb{N}} \def\PP{\mathbb{P}} \def\QQ{\mathbb{Q}} \def\RR{\mathbb{R}} \def\R{\mathbb{R}} \def\Z{\mathbb{Z}} \def\ZZ{\mathbb{Z}} \def\binom#1#2{{#1 \choose #2}} \def\mymod#1{\,(\mbox{mod}\,\, #1)} \def\pmatrix#1{\left( \begin{array}{ccc} #1 \end{array} \right)} %Structural macros \def\head#1{{\bf \noindent \newline #1: } \nopagebreak} \def\map#1#2#3{#1\!: #2\to #3} \def\benum{\begin{enumerate}} \def\eenum{\end{enumerate}} \def\soln{\head{Solution}} \def\first{\head{First Solution}} \def\second{{\bf \noindent \newline Second Solution: } \nopagebreak} \def\third{{\bf \noindent \newline Third Solution: } \nopagebreak} \def\note{{\bf \noindent \newline Note: } \nopagebreak} \def\be{\begin{enumerate}} \def\ee{\end{enumerate}} \def\beq{\begin{equation}} \def\eeq{\end{equation}} \def\beqa{\begin{eqnarray*}} \def\eeqa{\end{eqnarray*}} \def\bea{\begin{array}} \def\eea{\end{array}} \def\ii{\item} \def\ds{\dsp} \def\seqa{a_{1}, \dots, a_{n}} \def\seqb{b_{1}, \dots, b_{n}} \def\seqx{x_{1}, \dots, x_{n}} \def\lf{\lfloor} \def\rf{\rfloor} \def\Lp{\left(} \def\Rp{\right)} \def\L[{\left[} \def\R]{\right]} \def\alf{\alpha} \def\lcm{\mbox{lcm}} \def\reseteq{\setcounter{equation}{0}} \def\Avec{{(A_{1}, \ldots, A_{n})}} \def\Acup{{A_{1} \cup A_{2} \cup \cdots \cup A_{n}}} \def\crn{{\sqrt[3]{n}}} \def\ang{\angle} \def\ep{\epsilon} \def\dsp{\displaystyle} \def\oar{\overrightarrow} \newtheorem{lemma}{Lemma} %\newcounter{felist} %\def\bfe{\begin{list}{\alph{felist}}{\usecounter{fe} \def\bi{\begin{itemize}} \def\ei{\end{itemize}} \def\dd{\dots} \def\half{\frac{1}{2}} \def\sang{\sin \angle} \def\binom#1#2{{#1 \choose #2}} \def\bZ{\mathbb{Z}} \def\head#1{\begin{center} {\bf #1} \end{center}} \pagestyle{empty} \begin{document} %\input epsf \begin{center} ${\bf 29^{th}}$ {\bf United States of America Mathematical Olympiad} \end{center} \begin{center} {\bf Part I \hspace{ 6mm} 9 a.m. -12 noon} \end{center} \begin{center} {\bf May 2, 2000} \end{center} \bigskip \be \ii %PNN1 Call a real-valued function $f$ {\em very convex} if \[ \frac{f(x) + f(y)}{2} \ge f\Lp \frac{x+y}{2} \Rp + |x-y| \] holds for all real numbers $x$ and $y$. Prove that no very convex function exists. \vspace{5mm} \ii %AND1 Let $S$ be the set of all triangles $ABC$ for which \[ 5\Lp \frac{1}{AP} + \frac{1}{BQ} + \frac{1}{CR} \Rp - \frac{3}{\min\{ AP, BQ, CR \}} = \frac{6}{r}, \] where $r$ is the inradius and $P, Q, R$ are the points of tangency of the incircle with sides $AB, BC, CA,$ respectively. Prove that all triangles in $S$ are isosceles and similar to one another. \vspace{5mm} \ii %Stong3 A game of solitaire is played with $R$ red cards, $W$ white cards, and $B$ blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a blue card, then he is charged a penalty which is the number of white cards still in his hand. If he plays a white card, then he is charged a penalty which is twice the number of red cards still in his hand. If he plays a red card, then he is charged a penalty which is three times the number of blue cards still in his hand. Find, as a function of $R, W,$ and $B,$ the minimal total penalty a player can amass and all the ways in which this minimum can be achieved. \ee \vspace{80 mm} {\small \begin{center} Copyright \copyright \ \ Committee on the American Mathematics Competitions,\\ Mathematical Association of America \end{center} } \newpage \begin{center} ${\bf 29^{th}}$ {\bf United States of America Mathematical Olympiad} \end{center} \begin{center} {\bf Part II \hspace{ 6mm} 1 p.m. - 4 p.m.} \end{center} \begin{center} {\bf May 2, 2000} \end{center} \bigskip \be \ii[4.] %SOI4 Find the smallest positive integer $n$ such that if $n$ squares of a $1000 \times 1000$ chessboard are colored, then there will exist three colored squares whose centers form a right triangle with sides parallel to the edges of the board. \vspace{5mm} \ii[5.] %KED2 Let $A_1A_2A_3$ be a triangle and let $\omega_1$ be a circle in its plane passing through $A_1$ and $A_2.$ Suppose there exist circles $\omega_2, \omega_3, \dots, \omega_7$ such that for $k = 2, 3, \dots, 7,$ $\omega_k$ is externally tangent to $\omega_{k-1}$ and passes through $A_k$ and $A_{k+1},$ where $A_{n+3} = A_{n}$ for all $n \ge 1$. Prove that $\omega_7 = \omega_1.$ \vspace{5mm} \ii[6.] %AND3 Let $a_1, b_1, a_2, b_2, \dots , a_n, b_n$ be nonnegative real numbers. Prove that \[ \sum_{i, j = 1}^{n} \min\{a_ia_j, b_ib_j\} \le \sum_{i, j = 1}^{n} \min\{a_ib_j, a_jb_i\}. \] \ee \vspace{120mm} {\small \begin{center} Copyright \copyright \ \ Committee on the American Mathematics Competitions,\\ Mathematical Association of America \end{center} } \end{document}