Conference rusure::math

1782.0. "imo 93" by HERON::BUCHANAN (The was not found.) Mon Aug 16 1993 13:07

	The questions, followed by the scores, followed by a LATEX version of
the questions from a second source.

-------------------------------------------------------------------------------

Article 49779 of sci.math:
Newsgroups: sci.math
Path: nntpd.lkg.dec.com!news.crl.dec.com!pa.dec.com!decwrl!uunet!munnari.oz.au!hp9000.csc.cuhk.hk!uxmail!uxmail.ust.hk!eetszmei
From: [email protected] (Tsz-Mei Ko)
Subject: 34th IMO problems/results
Message-ID: <[email protected]>
Sender: [email protected] (usenet account)
Organization: Hong Kong University of Science and Technology
Date: Wed, 4 Aug 1993 08:37:00 GMT
Lines: 117


Problems for the 34th International Math Olympiad:
 (held in Istanbul, Turkey, July 17-24, 1993)

(For problems 1-3, the English version given out to the contestants
 are listed. For problems 4-6, the original proposed version are listed
 since I don't have the version given out to the contestants. Also,
 the following problems are slightly modified to be readable in ASCII.)

First Day (Time allowed is four and a half hours):

Problem 1 (proposed by IRE)
   Let f(x) = x^n + 5x^{n-1} + 3 where n>1 is an integer.
   Prove that f(x) cannot be expressed as the product of two polynomials,
   each of which has all its coefficients integers and degree at least 1.

Problem 2 (proposed by UNK)
   Let D be a point inside the acute-angled triangle ABC such that
          \angle ADB = \angle ACB + 90 degrees
   and       AC x BD = AD x BC.
   (a) Calculate the value of the ratio (AB x CD)/(AC x BD).
   (b) Prove that the tangents at C to the circumcircles of the
       triangles ACD and BCD are perpendicular.

Problem 3 (proposed by FIN)
   On an infinite chessboard, a game is played as follows.
   At the start, n^2 pieces are arranged on the chessboard in
   an n x n block of adjoining squares, one piece in each square.
   A move in the game is a jump in a horizontal or vertical direction
   over an adjacent occupied square to an unoccupied square immediately
   beyond. The piece which has been jumped over is then removed.
   Find those values of n for which the game can end with only one
   piece remaining on the board.

Second Day (Time allowed is four and a half hours):

Problem 4 (proposed by MAK)
   For three points A,B,C in the plane we define m(ABC) to be the smallest
   length of the three heights of the triangle ABC, where in the case
   A,B,C are collinear, m(ABC)=0. Let A,B,C be given points in the plane.
   Prove that for any X in the plane,
       m(ABC) is less than or equal to m(ABX) + m(AXC) + m(XBC).

Problem 5 (proposed by FRG)
   Let N = {1,2,3,...}. Determine if there exists a strictly increasing
   function f: N->N with the properties (1), (2):
   (1)    f(1) = 2
   (2)    f(f(n)) = f(n) + n   (n \in N).

Problem 6 (proposed by NET)
   Let n be an integer > 1. In a circular arrangement of n lamps
   L_0, ... L_{n-1}, each one of which can be either ON or OFF, we start
   with the situation where all lamps are ON, and then carry out a sequence
   of steps, Step_0, Step_1, .... If L_{j-1} (j is taken mod n) is ON,
   then Step_j changes the status of L_j (it goes from ON to OFF or from
   OFF to ON) but does not change the status of any of the other lamps.
   If L_{j-1} is OFF then Step_j does not change anything at all. Show that:

   (i) There is a positive integer M(n) such that after M(n) steps all
       lamps are ON again.
  (ii) If n has the form 2^k, then all lamps are ON after n^2 - 1 steps,
 (iii) If n has the form 2^k + 1, then all lamps are ON after n^2 - n + 1 steps.

------------------------------------------------------------------------------
Result of the 34th International Math Olympiad
 (held in Istanbul, Turkey, July 17-24, 1993)

rank country score    rank country score    rank country score
  1    CHN    215      26    COL     79      51    POR     35
  2    FRG    189      26    GEO     79      52    AZB/5   33
  3    BUL    178      28    ARM     78      52    FIN     33
  4    RUS    177      28    POL     78      52    PHI     33
  5    ROC    162      30    SIN     75      55    CRO     32
  6    IRA    153      31    LVA     73      56    EST     31
  7    USA    151      32    DEN     72      56    MON     31
  8    HUN    143      33    HKG     70      58    RTT     30
  9    VIE    138      34    BRA     60      58    RSA     30
 10    CZE    132      35    NET     58      60    MLD     29
 11    ROM    128      36    CUB     56      61    KRG/5   28
 12    SVK    126      37    BEL     55      62    MAC     24
 13    AUS    125      38    BLR/4   54      62    MEX     24
 14    UNK    118      39    SWE     51      64    ICE/4   23
 15    IND    116      40    MAR     49      65    LUX/1   20
 15    ROK    116      41    THA     47      66    ALB     18
 17    FRA    115      42    ARG     46      67    NCY     17
 18    CAN    113      42    SWT/4   46      68    BRN     16
 19    ISA    113      44    NOR/5   44      68    KUW     16
 20    JPN     98      45    ESP     43      70    INA     15
 21    UKR     96      45    NZL     43      71    BSN/2   14
 22    AUT     87      45    SLO/5   43      72    ALG/5    9
 23    ITA     86      48    MAK/4   42      72    TRK/2    9
 24    TUR     81      49    LIT     41 
 25    KAZ     80      50    IRE     39

Remarks:1. A full team consists of 6 students.
           BLR/4 means that the country BLR sent only 4 students.
        2. The maximum possible score is 252 points:
           6 students x 6 problems/student x 7 points/problems = 252 points.

Some statistics (from the IMO committee):
For teams: Number of participating teams: 73
           Mean score: 72.15
           Standard Error: 5.83
           Median: 58.00
           Standard Deviation: 49.78

For contestants: Number of participating contestants: 412
                 Mean score: 12.66
                 Standard Error: 0.48
                 Median: 10.00
                 Standard Deviation: 9.77

Future host countries for the IMO:  1994: Hong Kong (July 12-19, 1994)
                                    1995: Canada
                                    1996: India
                                    1997: Argentina
                                    1998: Taiwan



Article 50512 of sci.math:
Xref: nntpd.lkg.dec.com fj.sci.math:2183 fj.sources:2178 sci.math:50512
Newsgroups: fj.sci.math,fj.sources,sci.math
Path: nntpd.lkg.dec.com!pa.dec.com!decwrl!uunet!ccut!news.u-tokyo.ac.jp!s.u-tokyo!ismspc6!maruyama
From: [email protected] (MARUYAMA Naomasa)
Subject: 34th International Math. Oylmpiad (English Version)
Followup-To: fj.sci.math
Date: Thu, 12 Aug 1993 08:13:45 GMT
Nntp-Posting-Host: sunnm
Reply-To: [email protected]
Organization: The Inst. of Statistical Mathematics, Tokyo Japan
Sender: [email protected]
Message-ID: <[email protected]>
Lines: 146

Here is the LaTeX text of the English Version of 34th IMO problems.
Assumes A4 standard paper. Please enjoy it.

This is the text which I made after the contest. In the contest typesetted
version prepared by the organizing committee was used.

Naomasa MARUYAMA (Deputy leader of IMO34 Japan Team)

\documentstyle[12pt]{article}

%% A4 Page style by Hideki ISOZAKI ([email protected])

\oddsidemargin -0.54cm \evensidemargin -0.54cm
\topmargin -2cm \headheight 1pc \headsep 2pc
\footheight 1pc \footskip 2pc
\textheight 60pc \textwidth 40pc \columnsep 2pc \columnseprule 0pt

\renewcommand{\thefootnote}{\fnsymbol{footnote}}

\title{34th International Mathematical Olympiad\footnotemark \\
       Istanbul, 1993}

\footnotetext{Original copyright \copyright 1993 by IMO93 Problem Committee}
\footnotetext{This {\LaTeX} version
  is made by Mathematical Olympiad Foundation of Japan}
\footnotetext{Copyright \copyright 1993 by Mathematical Olympiad
  Foundation of Japan}
\date{}
\author{}
\pagestyle{empty}
\begin{document}
\maketitle\thispagestyle{empty}
%\maketitle
\newcommand{\IND}{\hspace*{3cm}}

Version: {\bf English} \hfill July 18, 1993\\

\begin{Large}
\begin{center}
{\bf FIRST DAY}
\end{center}
\end{Large}

\begin{itemize}
\item[1.] Let $f(x)=x^n+5x^{n-1}+3$ where $n>1$ is an integer. \\
      Prove that $f(x)$ cannot be expressed as the product of two polynomials,
      each of which has all its coefficients integers and degree at least 1.

\item[2.] Let D be a point inside the acute-angled triangle ABC such that
      \begin{description}
        \item \IND ${\rm \angle{ADB} = \angle{ACB} + 90^\circ }$
      \end{description}
      and
      \begin{description}
        \item \IND ${\rm AC \cdot BD = AD \cdot BC }$.
      \end{description}
      \begin{description}
        \item[(a)] Calculate the value of the ratio
              ${\rm \frac{AB \cdot CD}{AC \cdot BD} }$.
        \item[(b)] Prove that the tangents at C to the circumcircles
              of the triangles ACD and BCD are perpendicular.
      \end{description}

\item[3.] On an infinite chessboard, a game is played as follows.  \\
      At the start, $n^2$ pieces are arranged on the chessboard in an
      $n \times n$ block of adjoining squares, one piece in each square.
      A move in the game is a jump in a horizontal or vertical direction
      over an adjacent occupied square to an unoccupied square immediately
      beyond. The piece which has been jumped over is then removed. \\
      Find those values of $n$ for which the game can end with only one
      piece remaining on the board.

\end{itemize}

\begin{flushleft}
Each question is worth 7 points. \\
Time allowed is 4 1/2 hours.
\end{flushleft}

\newpage

Version: {\bf English} \hfill July 19, 1993 \\

\begin{Large}
\begin{center}
{\bf SECOND DAY}
\end{center}
\end{Large}

\begin{itemize}

\item[4.] For three points P,Q,R in the plane, we define $m({\rm PQR})$ 
      to be the minimum of the lengths of the altitudes of the triangle
      PQR (where $m({\rm PQR})=0$ when P,Q,R are collinear). \\
      Let A,B,C be given points in the plane. Prove that, for any point X
      in the plane,
      \begin{description}
        \item \IND $m({\rm ABC}) \leq m({\rm ABX}) + m({\rm AXC}) + m({\rm XBC})$.
      \end{description}

\item[5.] Let ${\bf N}=\{1,2,3,\cdots\}$.  \\
      Determine whether or not there exists a function
      $f: {\bf N}\rightarrow {\bf N}$ such that \\ \\
      \IND \begin{tabular}{lrcll}
             & $f(1)$    & $=$ & $2$,     &                         \\
             & $f(f(n))$ & $=$ & $f(n)+n$ & for all $n \in {\bf N}$ \\
         and &    $f(n)$ & $<$ & $f(n+1)$ & for all $n \in {\bf N}$. \\
           \end{tabular}

\item[6.] Let $n>1$ be an integer. There are $n$ lamps $L_0,L_1,\cdots,L_{n-1}$
      arranged in a circle. Each lamp is either ON or OFF. A sequence of steps
      $S_0,S_1,\cdots,S_i,\cdots$ is carried out. Step $S_j$ affects the
      state of $L_j$ only (leaving the state of all other lamps unaltered)
      as follows:
      \begin{description}
        \item \hspace*{1em} if $L_{j-1}$ is ON, $S_j$ changes the state of $L_j$ from
           ON to OFF or from OFF to ON;
        \item \hspace*{1em} if $L_{j-1}$ is OFF, $S_j$ leaves the state of $L_j$ unchanged.
      \end{description}
      The lamps are labelled mod n, that is,
      \begin{description}
        \item \IND $L_{-1}=L_{n-1},L_0=L_n,L_1=L_{n+1}$, etc. .
      \end{description}
      Initially all lamps are ON. Show that
      \begin{itemize}
      \item[(a)] there is a positive integer $M(n)$ such that after $M(n)$
            steps all the lamps are ON again;
      \item[(b)] if $n$ has the form $2^k$ then all the lamps are ON after
            $n^2-1$ steps;
      \item[(c)] if $n$ has the form $2^k+1$ then all the lamps are ON after
            $n^2-n+1$ steps.
      \end{itemize}
\end{itemize}

\begin{flushleft}
Each question is worth 7 points. \\
Time allowed is 4 1/2 hours.
\end{flushleft}

\end{document}

--
1993年08月12日(木)
--------
丸山彫苳殺苳算＠統計数理研瘢雹究所
[email protected]