-- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 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 --------------------- 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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