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By Arnfried Kemnitz

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If the probability that both marbles are white is m/n, where m and n are relatively prime, find m + n. 6. How many pairs of positive integers m, n have n < m < 1000000 and their arithmetic mean equal to their geometric mean plus 2? 7. x, y, z are positive reals such that xyz = 1, x + 1/z = 5, y + 1/x = 29. Find z + 1/y. 8. A sealed conical vessel is in the shape of a right circular cone with height 12, and base radius 5. The vessel contains some liquid. When it is held point down with the base horizontal the liquid is 9 deep.

A group of 1000 switches are all at position A. Each switch has a unique label 2a3b5c, where a, b, c = 0, 1, 2, ... , or 9. A 1000 step process is now carried out. At each step a different switch S is taken and all switches whose labels divide the label of S are turned one place. For example, if S was 2·3·5, then the 8 switches with labels 1, 2, 3, 5, 6, 10, 15, 30 would each be turned one place. How many switches are in position A after the process has been completed? 8. T is the region of the plane x + y + z = 1 with x,y,z ≥0.

9. Given a lattice of regular hexagons. A bug crawls from vertex A to vertex B along the edges of the hexagons, taking the shortest possible path (or one of them). Prove that it travels a distance at least AB/2 in one direction. If it travels exactly AB/2 in one direction, how many edges does it traverse? 10. A circle center O is inscribed in ABCD (touching every side). Prove that ∠ AOB + ∠ COD = 180o. 11. The natural numbers a, b, n are such that for every natural number k not equal to b, b - k divides a - kn.

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