1/√h ≥ 1/√AD + 1/√BC.
| A1. Prove that the set {1, 2, ... , 1989} can be expressed as the disjoint union of subsets A1, A2, ... , A117 in such a way that each Ai contains 17 elements and the sum of the elements in each Ai is the same. |
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| A2. In an acute-angled triangle ABC, the internal bisector of angle A meets the circumcircle again at A1. Points B1 and C1 are defined similarly. Let A0 be the point of intersection of the line AA1 with the external bisectors of angles B and C. Points B0 and C0 are defined similarly. Prove that the area of the triangle A0B0C0 is twice the area of the hexagon AC1BA1CB1 and at least four times the area of the triangle ABC. |
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| A3. Let n and k be positive integers, and let S be a set of n points in the plane such that no three points of S are collinear, and for any point P of S there are at least k points of S equidistant from P. Prove that k < 1/2 + √(2n). |
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B1. Let ABCD be a convex quadrilateral such that the sides AB, AD, BC satisfy AB = AD + BC. There exists a point P inside the quadrilateral at a distance h from the line CD such that AP = h + AD and BP = h + BC. Show that:
1/√h ≥ 1/√AD + 1/√BC. |
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| B2. Prove that for each positive integer n there exist n consecutive positive integers none of which is a prime or a prime power. |
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| B3. A permutation {x1, x2, ... , xm} of the set {1, 2, ... , 2n} where n is a positive integer is said to have property P if |xi - xi+1| = n for at least one i in {1, 2, ... , 2n-1}. Show that for each n there are more permutations with property P than without. |
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