From - Tue Aug 12 17:17:33 1997
From: jpf@hydra.cfm.brown.edu (Jim P. Ferry)
Newsgroups: sci.math
Subject: IMO 1997 - The Problems
Date: 1 Aug 1997 22:45:47 GMT
Organization: Brown University Center for Fluid Mechanics
Message-ID: <5rtour$t4f@cocoa.brown.edu>

In article <5rfbpd$8ej$1@news.duke.edu>, dkrain@cauchy.math.duke.edu (David Kraines) writes:

|> Exam is included in Argentine web site
|> http://www.oma.org.ar/imo/imo-engl.html

Here are the problems, downloaded from the above web site and slightly altered so
as to be readable as a pure text file.  The site also contains individual results,
but not team results (which I'd be interested in seeing).

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38th IMO:  First day
Mar del Plata, Argentina - July 24, 1997


---------- Problem 1 ----------

In the plane the points with integer coordinates are the vertices of unit squares.
The squares are coloured alternately black and white (as on a chessboard). 

For any pair of positive integers m and n, consider a right-angled triangle whose 
vertices have integer coordinates and whose legs, of lengths m and n, lie along 
edges of the squares.

Let S1 be the total area of the black part of the triangle and S2 be the total
area of the white part. Let

f(m,n) = | S1 - S2 |.

(a) Calculate f(m,n) for all positive integers m and n which are either both even
    or both odd. 
                           1
(b) Prove that  f(m,n) <=  - max(m,n)  for all m and n.
                           2

(c) Show that there is no constant C such that f(m,n) < C for all m and n. 


---------- Problem 2 ----------

Angle A is the smallest in the triangle ABC. 

The points B and C divide the circumcircle of the triangle into two arcs. Let U
be an interior point of the arc between B and C which does not contain A. 

The perpendicular bisectors of AB and AC meet the line AU at V and W,
respectively.  The lines BV and CW meet at T. 

Show that 

AU = TB + TC.


---------- Problem 3 ----------

Let x_1, x_2, ..., x_n be real numbers satisfying the conditions: 

    | x_1 + x_2 + ... + x_n | = 1 

and 
             n+1
    |x_i| <= ---  for i = 1, 2, ..., n.
              2 

Show that there exists a permutation  y_1, y_2, ..., y_n  of  x_1, x_2, ..., x_n 
such that
                                 n+1
| y_1 + 2 y_2 + ... + n y_n | <= --- .
                                  2

-------------------------------
Each problem is worth 7 points.
Time: 4 1/2 hours.

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38th IMO:  Second day
Mar del Plata, Argentina - July 25, 1997


---------- Problem 4 ----------

An n x n matrix (square array) whose entries come from the set S = {1, 2, ...,
2n-1}, is called a silver matrix if, for each i = 1, ..., n, the i_th row and the 
i_th column together contain all elements of S.  Show that

(a) there is no silver matrix for n = 1997;

(b) silver matrices exist for infinitely many values of n.


---------- Problem 5 ----------

Find all pairs (a,b) of integers a>=1, b>=1 that satisfy the equation

   2
 (b )    a
a     = b .


---------- Problem 6 ----------

For each positive integer n, let f(n) denote the number of ways of representing
n as a sum of powers of 2 with nonnegative integer exponents.

Representations which differ only in the ordering of their summands are considered
to be the same.  For instance, f(4)=4, because the number 4 can be represented in 
the following four ways:

4; 2+2; 2+1+1; 1+1+1+1.

Prove that, for any integer n>=3:

   2                  2
 (n / 4)      n     (n / 2)
2        < f(2 ) < 2       .


-------------------------------
Each problem is worth 7 points.
Time: 4 1/2 hours.

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