| 1. A licence plate is 3 letters followed by 3 digits. If all possible licence plates are equally likely, what is the probability that a plate has either a letter palindrome or a digit palindrome (or both)? |
| 2. 20 equal circles are packed in honeycomb fashion in a rectangle. The outer rows have 7 circles, and the middle row has 6. The outer circles touch the sides of the rectangle. Find the long side of the rectangle divided by the short side. |
| 3. Jane is 25. Dick's age is d > 25. In n years both will have two-digit ages which are obtained by transposing digits (so if Jane will be 36, Dick will be 63). How many possible pairs (d, n) are there? |
| 4. The sequence x1, x2, x3, ... is defined by xk = 1/(k2 + k). A sum of consecutive terms xm + xm+1 + ... + xn = 1/29. Find m and n. |
| 5. D is a regular 12-gon. How many squares (in the plane of D) have two or more of their vertices as vertices of D? |
| 6. The solutions to log225x + log64y = 4, logx225 - logy64 = 1 are (x, y) = (x1, y1) and (x2, y2). Find log30(x1y1x2y2). |
| 7. What are the first three digits after the decimal point in (102002 + 1)10/7? You may use the extended binomial theorem: (x + y)r = xr(1 + 4 (y/x) + r(r-1)/2! (y/x)2 + r(r-1)(r-2)/3! (y/x)3 + ...) for r real and |x/y| < 1 |
| 8. Find the smallest integer k for which there is more than one non-decreasing sequence of positive integers a1, a2, a3, ... such that a9 = k and an+2 = an+1 + an. |
| 9. A, B, C paint a long line of fence-posts. A paints the first, then every ath, B paints the second then every bth, C paints the third, then every cth. Every post gets painted just once. Find all possible triples (a, b, c). |
| 10. ABC is a triangle with angle B = 90o. AD is an angle bisector. E lies on the side AB with AE = 3, EB = 9, and F lies on the side AC with AF = 10, FC = 27. EF meets AD at G. Find the nearest integer to area GDCF. |
| 11. A cube with two faces ABCD, BCEF, has side 12. The point P is on the face BCEF a perpendicular distance 5 from the edge BC and from the edge CE. A beam of light leaves A and travels along AP, at P it is reflected inside the cube. Each time it strikes a face it is reflected. How far does it travel before it hits a vertex? |
| 12. The complex sequence z0, z1, z2, ... is defined by z0 = i + 1/137 and zn+1 = (zn + i)/(zn - i). Find z2002. |
| 13. The triangle ABC has AB = 24. The median CE is extended to meet the circumcircle at F. CE = 27, and the median AD = 18. Find area ABF. |
| 14. S is a set of positive integers containing 1 and 2002. No elements are larger than 2002. For every n in S, the arithmetic mean of the other elements of S is an integer. What is the largest possible number of elements of S? |
| 15. ABCDEFGH is a polyhedron. Face ABCD is a square side 12. Face ABFG is a trapezoid with GF parallel to AB and GF = 6, AG = BF = 8. Face CDE is an isosceles triangle with ED = EC = 14. E is a distance 12 from the plane ABCD. The other faces are EFG, ADEG and BCEF. Find EG2. |
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 4 Aug 03