| 1. The product N of three positive integers is 6 times their sum. One of the integers is the sum of the other two. Find the sum of all possible values of N. |
| 2. N is the largest multiple of 8 which has no two digits the same. What is N mod 1000? |
| 3. How many 7-letter sequences are there which use only A, B, C (and not necessarily all of those), with A never immediately followed by B, B never immediately followed by C, and C never immediately followed by A? |
| 4. T is a regular tetrahedron. T' is the tetrahedron whose vertices are the midpoints of the faces of T. Find vol T'/vol T. |
| 5. A log is in the shape of a right circular cylinder diameter 12. Two plane cuts are made, the first perpendicular to the axis of the log and the second at a 45o angle to the first, so that the line of intersection of the two planes touches the log at a single point. The two cuts remove a wedge from the log. Find its volume. |
| 6. A triangle has sides 13, 14, 15. It is rotated through 180o about its centroid to form an overlapping triangle. Find the area of the union of the two triangles. |
| 7. ABCD is a rhombus. The circumradii of ABD, ACD are 12.5, 25. Find the area of the rhombus. |
| 8. Corresponding terms of two arithmetic progressions are multiplied to give the sequence 1440, 1716, 1848, ... . Find the eighth term. |
| 9. The roots of x4 - x3 - x2 - 1 = 0 are a, b, c, d. Find p(a) + p(b) + p(c) + p(d), where p(x) = x6 - x5 - x3 - x2 - x. |
| 10. Find the largest possible integer n such that √n + √(n+60) = √m for some non-square integer m. |
| 11. ABC has AC = 7, BC = 24, angle C = 90o. M is the midpoint of AB, D lies on the same side of AB as C and had DA = DB = 15. Find area CDM. |
| 12. n people vote for one of 27 candidates. Each candidate's percentage of the vote is at least 1 less than his number of votes. What is the smallest possible value of n? (So if a candidate gets m votes, then 100m/n ≤ m-1.) |
| 13. A bug moves around a wire triangle. At each vertex it has 1/2 chance of moving towards each of the other two vertices. What is the probability that after crawling along 10 edges it reaches its starting point? |
| 14. ABCDEF is a convex hexagon with all sides equal and opposite sides parallel. Angle FAB = 120o. The y-coordinates of A, B are 0, 2 respectively, and the y-coordinates of the other vertices are 4, 6, 8, 10 in some order. Find its area. |
| 15. The distinct roots of the polynomial x47 + 2x46 + 3x45 + ... + 24x24 + 23x23 + 22x22 + ... + 2x2 + x are z1, z2, ... , zn. Let zk2 have imaginary part bki. Find |b1| + |b2| + ... + |bn|. |
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
4 Aug 2003
Last updated/corrected 15 Mar 04