# sample problems

Chuck Biehl (bieh9435@dpnet.net)
Mon, 25 Aug 1997 16:35:03 -0400

I couldn't resist passing this along as an example of a humbling
experience...
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L. Charles (Chuck) Biehl
Mathematics, Science & Technology
The Charter School of Wilmington
(v)302-651-2727  (f)302-652-1246
http://dimacs.rutgers.edu/~biehl

Subject:
Fwd: IMO
Date:
Sun, 24 Aug 1997 14:48:49 -0400 (EDT)
From:
KarenDM@aol.com
To:
cpamlist@cedar.cic.net

Dear Friends,

I thought I'd send you the questions that my student Nathan Curtis, on
our
International Mathematics Olympiad team, just sent me.  We placed fourth
in
the world tied with Russia.  Nathan was one of the gold medal winners.

Cheers,
Karen Dee Michalowicz
Virginia, l994 Secondary Math
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Forwarded message:
Subj:    IMO
Date:    97-08-23 22:23:57 EDT
From:    TJ Nate
To:      KarenDM

Hi, Mrs. Mikey,

I finally got AOL up on my computer again.  Here are the questions, or
at
least an equivalent paraphrasing of the questions:

Day 1:

1. The lattice points in the plane (points w/integer coordinates) are
the
vertices of unit squares tiling the plane, and these squares are
alternately
colored black and white, as on a chessboard.  Given positive integers m
and
n, consider a right triangle with legs of length m and n along the x-
and
y-axes (or any other horizontal and vertical "lattice line").  Let the
area
of the black regions of the triangle be S1, let the area of the white
regions
be S2, and let f(m,n)=abs(S1-S2).

a. Calculate f(m,n) for the cases when m and n are both odd or both
even.

b. Prove that f(m,n)<=1/2 max(m,n), for all m and n (<= is less or equal
to).

c. Prove that there is no constant C s.t. f(m,n)<C for all m, n.

2. In triangle ABC, let A be the smallest angle, and let U be an
arbitrary
point on the minor arc BC of the circumcircle of ABC.  Let the
perpendicular
bisectors of AB and AC meet AU at V and W, respectively, and let lines
BU
and
CV intersect at T.  Prove that  AU = BT + CT.

3. X1, X2,..., Xn are real numbers satisfying
a) abs(X1+X2+...+Xn)=1, and
b) abs(Xi)<=(n+1)/2, for i=1 to n.
Prove that there exists some permutation Y1, Y2,..., Yn of the Xi's such
that
abs(Y1+2Y2+3Y3+...+nYn)<=(n+1)/2.

Day 2

4. An n by n matrix of elements from the set S={1, 2,..., 2n-1} is
called a
silver matrix iff for every i from 1 to n, the i-th row and i-th column
together contain all the elements of S.

a. Prove that no silver matrices exist for n=1997, and

b. Prove that silver matrices exist for infinitely many values of n.

5. Find all solutions (a, b) in positive integers to the equation
a^(b^2) = b^a.

6. For a positive integer n, let f(n) denote the number of ways n can be
written as the sum of powers of two, not including different
permutations.
For example, f(4)=4, since 4=2+2=2+1+1=1+1+1+1 are the four different
ways
of writing 4 as the sum of powers of two.  Prove that
2^((n^2)/4) < f(2^n) < 2((n^2)/2),
for all n>=3.

I got seven's on the first five, and a five on number six (only ten
people
got a seven worldwide on that one).  Given an extra five minutes, I
probably
would have gotten a six, since a point was taken off for typos
("mathos",
rather) in my paper (I was writing stuff up in the final minutes, and
the
sound of other students putting their answer sheets in their respective
envelopes was really distracting, so I was very nervous at the time.)
Most
of
my mistakes were in writing some  expressions in n-1 rather than n, when
I
was trying to induct, thus getting mismatched sides which worked out in
my
scrap work.  Fortunately, scrap work counts at the IMO, so I didn't lose
more
points.

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