\documentclass[11pt]{report} %\input{pjtamc} \pagestyle{empty} \textwidth 17cm \textheight 21cm \parindent=0pt \parskip=5pt \hoffset -2cm %\voffset 1cm \def\<{\angle} \def\sol{{\bf Solution.}\\} \begin{document} \begin{center} \vskip 3cm {\Large\bf XIII Asian Pacific Mathematics Olympiad\\ March, 2001} \end{center} \vskip 0.3cm {\em Time allowed: 4 hours} {\em No calculators to be used} {\em Each question is worth 7 points} \vskip0.3cm {\bf Problem 1.} For a positive integer $n$ let $S(n)$ be the sum of digits in the decimal representation of $n$. Any positive integer obtained by removing several (at least one) digits from the right-hand end of the decimal representation of $n$ is called a $stump$ of $n$. Let $T(n)$ be the sum of all stumps of $n$. Prove that $n=S(n)+9T(n)$. \bigskip {\bf Problem 2.}\\ Find the largest positive integer $N$ so that the number of integers in the set $\{1,2,\dots,N\}$ which are divisible by 3 is equal to the number of integers which are divisible by 5 or 7 (or both). \bigskip {\bf Problem 3.}\\ Let two equal regular $n$-gons $S$ and $T$ be located in the plane such that their intersection is a $2n$-gon ($n\ge 3$). The sides of the polygon $S$ are coloured in red and the sides of $T$ in blue. \medskip Prove that the sum of the lengths of the blue sides of the polygon $S\cap T$ is equal to the sum of the lengths of its red sides. \bigskip {\bf Problem 4.}\\ A point in the plane with a cartesian coordinate system is called a {\em mixed point} if one of its coordinates is rational and the other one is irrational. Find all polynomials with real coefficients such that their graphs do not contain any mixed point. \bigskip {\bf Problem 5.}\\ Find the greatest integer $n$, such that there are $n+4$ points $A$, $B$, $C$, $D$, $X_1,\dots,~X_n$ in the plane with $AB\ne CD$ that satisfy the following condition: for each $i=1,2,\dots,n$ triangles $ABX_i$ and $CDX_i$ are equal. \end{document}