| 1. n has 4 digits, which are consecutive integers in decreasing order (from left to right). Find the sum of the possible remainders when n is divided by 37. |
| 2. The set A consists of m consecutive integers with sum 2m. The set B consists of 2m consecutive integers with sum m. The difference between the largest elements of A and B is 99. Find m. |
| 3. P is a convex polyhedron with 26 vertices, 60 edges and 36 faces. 24 of the faces are triangular and 12 are quadrilaterals. A space diagonal is a line segment connecting two vertices which do not belong to the same face. How many space diagonals does P have? |
| 4. A square X has side 2. S is the set of all segments length 2 with endpoints on adjacent sides of X. The midpoints of the segments in S enclose a region with area A. Find 100A to the nearest whole number. |
| 5. A and B took part in a two-day maths contest. At the end both had attempted questions worth 500 points. A scored 160 out of 300 attempted on the first day and 140 out of 200 attempted on the second day, so his two-day success ratio was 300/500 = 3/5. B's attempted figures were different from A's (but with the same two-day total). B had a positive integer score on each day. For each day B's success ratio was less than A's. What is the largest possible two-day success ratio that B could have achieved? |
| 6. An integer is snakelike if its decimal digits d1d2...dk satisfy di < di+1 for i odd and di > di+1 for i even. How many snakelike integers between 1000 and 9999 have four distinct digits? |
| 7. Find the coefficient of x2 in the polynomial (1-x)(1+2x)(1-3x)...(1+14x)(1-15x). |
| 8. A regular n-star is the union of n equal line segments P1P2, P2P3, ... , PnP1 in the plane such that the angles at Pi are all equal and the path P1P2...PnP1 turns counterclockwise through an angle less than 180o at each vertex. There are no regular 3-stars, 4-stars or 6-stars, but there are two non-similar regular 7-stars. How many non-similar regular 1000-stars are there? |
| 9. ABC is a triangle with sides 3, 4, 5 and DEFG is a 6 x 7 rectangle. A line divides ABC into a triangle T1 and a trapezoid R1. Another line divides the rectangle into a triangle T2 and a trapezoid R2, so that T1 and T2 are similar, and R1 and R2 are similar. Find the smallest possible value of area T1. |
| 10. A circle radius 1 is randomly placed so that it lies entirely inside a 15 x 36 rectangle ABCD. Find the probability that it does not meet the diagonal AC. |
| 11. The surface of a right circular cone is painted black. The cone has height 4 and its base has radius 3. It is cut into two parts by a plane parallel to the base, so that the volume of the top part (the small cone) divided by the volume of the bottom part (the frustrum) equals k and painted area of the top part divided by the painted are of the bottom part also equals k. Find k. |
| 12. Let S be the set of all points (x,y) such that x, y ∈ (0,1], and [log2(1/x)] and [log5(1/y)] are both even. Find area S. |
| 13. The roots of the polynomial (1 + x + x2 + ... + x17)2 - x17 are rk ei2πak, for k = 1, 2, ... , 34 where 0 < a1 ≤ a2 ≤ ... ≤ a34 < 1 and rk are positive. Find a1 + a2 + a3 + a4 + a5. |
| 14. A unicorn is tethered by a rope length 20 to the base of a cylindrical tower. The rope is attached to the tower at ground level and to the unicorn at height 4 and pulled tight. The unicorn's end of the rope is a distance 4 from the nearest point of the tower. Find the length of the rope which is in contact with the tower. |
| 15. Define f(1) = 1, f(n) = n/10 if n is a multiple of 10 and f(n) = n+1 otherwise. For each positive integer m define the sequence a1, a2, a3, ... by a1 = m, an+1 = f(an). Let g(m) be the smallest n such that an = 1. For example, g(100) = 3, g(87) = 7. Let N be the number of positive integers m such that g(m) = 20. How many distinct prime factors does N have? |
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
28 March 2004
Last updated/corrected 28 Mar 04