| 1. How many even integers between 4000 and 7000 have all digits different? |
| 2. Starting at the origin, an ant makes 40 moves. The nth move is a distance n2/2 units. Its moves are successively due E, N, W, S, E, N ... . How far from the origin does it end up? |
| 3. In a fish contest one contestant caught 15 fish. The other contestants all caught less. an contestants caught n fish, with a0 = 9, a1 = 5, a2 = 7, a3 = 23, a13 = 5, a14 = 2. Those who caught 3 or more fish averaged 6 fish each. Those who caught 12 or fewer fish averaged 5 fish each. What was the total number of fish caught in the contest? |
| 4. How many 4-tuples (a, b, c, d) satisfy 0 < a < b < c < d < 500, a + d = b + c, and bc - ad = 93? |
| 5. Let p0(x) = x3 + 313x2 - 77x - 8, and pn(x) = pn-1(x-n). What is the coefficient of x in p20(x)? |
| 6. What is the smallest positive integer that can be expressed as a sum of 9 consecutive integers, and as a sum of 10 consecutive integers, and as a sum of 11 consecutive integers? |
| 7. Six numbers are drawn at random, without replacement, from the set {1, 2, 3, ... , 1000}. Find the probability that a brick whose side lengths are the first three numbers can be placed inside a box with side lengths the second three numbers with the sides of the brick and the box parallel. |
| 8. S has 6 elements. How many ways can we select two (possibly identical) subsets of S whose union is S? |
| 9. Given 2000 points on a circle. Add labels 1, 2, ... , 1993 as follows. Label any point 1. Then count two points clockwise and label the point 2. Then count three points clockwise and label the point 3, and so on. Some points may get more than one label. What is the smallest label on the point labeled 1993? |
| 10. A polyhedron has 32 faces, each of which has 3 or 5 sides. At each of it s V vertices it has T triangles and P pentagons. What is the value of 100P + 10T + V? You may assume Euler's formula (V + F = E + 2, where F is the number of faces and E the number of edges). |
| 11. A and B play a game repeatedly. In each game players toss a fair coin alternately. The first to get a head wins. A starts in the first game, thereafter the loser starts the next game. Find the probability that A wins the sixth game. |
| 12. A = (0, 0), B = (0, 420), C = (560, 0). P1 is a point inside the triangle ABC. Pn is chosen at random from the midpoints of Pn-1A, Pn-1B, and Pn-1C. If P7 is (14, 92), find the coordinates of P1. |
| 13. L, L' are straight lines 200 ft apart. A and A' start 200 feet apart, A on L and A' on L'. A circular building 100 ft in diameter lies midway between the paths and the line joining A and A' touches the building. They begin walking in the same direction (past the building). A walks at 1 ft/sec, A' walks at 3 ft/sec. Find the amount of time before they can see each other again. |
| 14. R is a 6 x 8 rectangle. R' is another rectangle with one vertex on each side of R. R' can be rotated slightly and still remain within R. Find the smallest perimeter that R' can have. |
| 15. The triangle ABC has AB = 1995, BC = 1993, CA = 1994. CX is an altitude. Find the distance between the points at which the incircles of ACX and BCX touch CX. |
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
30 Sep 2003
Last updated/corrected 30 Sep 03