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A1. In the plane the points with integer coordinates are the vertices of unit squares. The squares are colored 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 the 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.
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| A2. The angle at A is the smallest angle 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. |
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| A3. Let x1, x2, ... , xn be real numbers satisfying |x1 + x2 + ... + xn| = 1 and |xi| ≤ (n+1)/2 for all i. Show that there exists a permutation yi of xi such that |y1 + 2y2 + ... + nyn| ≤ (n+1)/2. |
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B1. An n x n matrix whose entries come from the set S = {1, 2, ... , 2n-1} is called a silver matrix if, for each i = 1, 2, ... , n, the ith row and the ith column together contain all elements of S. Show that:
(a) there is no silver matrix for n = 1997;
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| B2. Find all pairs (a, b) of positive integers that satisfy ab2 = ba. |
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| B3. For each positive integer n, let f(n) denote the number of ways of representing n as a sum of powers of 2 with non-negative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For example, f(4) = 4, because 4 can be represented as 4, 2 + 2, 2 + 1 + 1 or 1 + 1 + 1 + 1. Prove that for any integer n ≥ 3, 2n2/4 < f(2n) < 2n2/2. |
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