| 1. Let x1 = 97, x2 = 2/x1, x3 = 3/x2, x4 = 4/x3, ... , x8 = 8/x7. Find x1x2 ... x8. |
| 2. The triangle ABC has angle B = 90o. When it is rotated about AB it gives a cone volume 800π. When it is rotated about BC it gives a cone volume 1920π. Find the length AC. |
| 3. m and n are positive integers such that N = (m + ni)3 - 107i is a positive integer. Find N. |
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4. ABCD is a square side 1. Points A', B', C', D' are taken on the sides AB, BC, CD, DA respectively so that AA'/AB = BB'/BC = CC'/CD = DD'/DA = 1/n. The strip bounded by the lines AC' and A'C meets the strip bounded by the lines BD' and B'D in a square area 1/1985. Find n.
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| 5. The integer sequence a1, a2, a3, ... satisfies an+2 = an+1 - an for n > 0. The sum of the first 1492 terms is 1985, and the sum of the first 1985 terms is 1492. Find the sum of the first 2001 terms. |
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6. A point is taken inside a triangle ABC and lines are drawn through the point from each vertex, thus dividing the triangle into 6 parts. Four of the parts have the areas shown. Find area ABC.
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| 7. The positive integers A, B, C, D satisfy A5 = B4, C3 = D2 and C = A + 19. Find D - B. |
| 8. Approximate each of the numbers 2.56, 2.61, 2.65, 2.71, 2.79, 2.82, 2.86 by integers, so that the 7 integers have the same sum and the maximum absolute error E is as small as possible. What is 100E? |
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9. Three parallel chords of a circle have lengths 2, 3, 4 and subtend angles x, y, x + y at the center (where x + y < 180o). Find cos x.
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| 10. How many of 1, 2, 3, ... , 1000 can be expressed in the form [2x] + [4x] + [6x] + [8x], for some real number x? |
| 11. The foci of an ellipse are at (9, 20) and (49, 55), and it touches the x-axis. What is the length of its major axis? |
| 12. A bug crawls along the edges of a regular tetrahedron ABCD with edges length 1. It starts at A and at each vertex chooses its next edge at random (so it has a 1/3 chance of going back along the edge it came on, and a 1/3 chance of going along each of the other two). Find the probability that after it has crawled a distance 7 it is again at A is p. |
| 13. Let f(n) be the greatest common divisor of 100 + n2 and 100 + (n+1)2 for n = 1, 2, 3, ... . What is the maximum value of f(n)? |
| 14. In a tournament each two players played each other once. Each player got 1 for a win, 1/2 for a draw, and 0 for a loss. Let S be the set of the 10 lowest-scoring players. It is found that every player got exactly half his total score playing against players in S. How many players were in the tournament? |
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15. A 12 x 12 square is divided into two pieces by joining to adjacent side midpoints. Copies of the triangular piece are placed on alternate edges of a regular hexagon and copies of the other piece are placed on the other edges. The resulting figure is then folded to give a polyhedron with 7 faces. What is the volume of the polyhedron?
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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
22 March 2003
Last updated/corrected 22 Mar 03