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A1. Consider two coplanar circles of radii R > r with the same center. Let P be a fixed point on the smaller circle and B a variable point on the larger circle. The line BP meets the larger circle again at C. The perpendicular to BP at P meets the smaller circle again at A (if it is tangent to the circle at P, then A = P).
(i) Find the set of values of AB2 + BC2 + CA2.
(ii) Find the locus of the midpoint of BC.
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A2. Let n be a positive integer and let A1, A2, ... , A2n+1 be subsets of a set B. Suppose that:
(i) Each Ai has exactly 2n elements,
(ii) The intersection of every two distinct Ai contains exactly one element, and
(iii) Every element of B belongs to at least two of the Ai.
For which values of n can one assign to every element of B one of the numbers 0 and 1 in such a way that each Ai has 0 assigned to exactly n of its elements?
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A3. A function f is defined on the positive integers by: f(1) = 1; f(3) = 3; f(2n) = f(n), f(4n + 1) = 2f(2n + 1) - f(n), and f(4n + 3) = 3f(2n + 1) - 2f(n) for all positive integers n. Determine the number of positive integers n less than or equal to 1988 for which f(n) = n.
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B1. Show that the set of real numbers x which satisfy the inequality:
1/(x - 1) + 2/(x - 2) + 3/(x - 3) + ... + 70/(x - 70) ≥ 5/4
is a union of disjoint intervals, the sum of whose lengths is 1988.
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B2. ABC is a triangle, right-angled at A, and D is the foot of the altitude from A. The straight line joining the incenters of the triangles ABD and ACD intersects the sides AB, AC at K, L respectively. Show that the area of the triangle ABC is at least twice the area of the triangle AKL.
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B3. Let a and b be positive integers such that ab + 1 divides a2 + b2. Show that (a2 + b2)/(ab + 1) is a perfect square.
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