| 1. Find the smallest a5, such that a1, a2, a3, a4, a5 is a strictly increasing arithmetic progression with all terms prime. |
| 2. A line through the origin divides the parallelogram with vertices (10, 45), (10, 114), (28, 153), (28, 84) into two congruent pieces. Find its slope. |
| 3. Find the sum of all positive integers n for which n2 - 19n + 99 is a perfect square. |
| 4. Two squares side 1 are placed so that their centers coincide. The area inside both squares is an octagon. One side of the octagon is 43/99. Find its area. |
| 5. For any positive integer n, let t(n) be the (non-negative) difference between the digit sums of n and n+2. For example t(199) = |19 - 3| = 16. How many possible values t(n) are less than 2000? |
| 6. A map T takes a point (x, y) in the first quadrant to the point (√x, √y). Q is the quadrilateral with vertices (900, 300), (1800, 600), (600, 1800), (300, 900). Find the greatest integer not exceeding the area of T(Q). |
| 7. A rotary switch has four positions A, B, C, D and can only be turned one way, so that it can be turned from A to B, from B to C, from C to D, or from D to A. 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. S is the set of points (a, b, c) in T such that just two of the following three inequalities hold: a ≤ 1/2, b ≤ 1/3, c ≤ 1/6. Find area S/area T. |
| 9. f is a complex-valued function on the complex numbers such that function f(z) = (a + bi)z, where a and b are real and |a + ib| = 8. It has the property that f(z) is always equidistant from 0 and z. Find b. |
| 10. S is a set of 10 points in the plane, no three collinear. There are 45 segments joining two points of S. Four distinct segments are chosen at random from the 45. Find the probability that three of these segments form a triangle (so they all involve two from the same three points in S). |
| 11. Find sin 5o + sin 10o + sin 15o + ... + sin 175o. You may express the answer as tan(a/b). |
| 12. The incircle of ABC touches AB at P and has radius 21. If AP = 23 and PB = 27, find the perimeter of ABC. |
| 13. 40 teams play a tournament. Each team plays every other team just once. Each game results in a win for one team. If each team has a 50% chance of winning each game, find the probability that at the end of the tournament every team has won a different number of games. |
| 14. P lies inside the triangle ABC, and angle PAB = angle PBC = angle PCA. If AB = 13, BC = 14, CA = 15, find tan PAB. |
| 15. A paper triangle has vertices (0, 0), (34, 0), (16, 24). The midpoint triangle has as its vertices the midpoints of the sides. The paper triangle is folded along the sides of its midpoint triangle to form a pyramid. What is the volume of the pyramid? |
To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
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© John Scholes
jscholes@kalva.demon.co.uk
17 Aug 2003
Last updated/corrected 8 Dec 03