| 1. Find 2/log4(20006) + 3/log5(20006). |
| 2. How many lattice points lie on the hyperbola x2 - y2 = 20002? |
| 3. A deck of 40 cards has four each of cards marked 1, 2, 3, ... 10. Two cards with the same number are removed from the deck. Find the probability that two cards randomly selected from the remaining 38 have the same number as each other. |
| 4. What is the smallest positive integer with 12 positive even divisors and 6 positive odd divisors? |
| 5. You have 8 different rings. Let n be the number of possible arrangements of 5 rings on the four fingers of one hand (each finger has zero or more rings, and the order matters). Find the three leftmost non-zero digits of n. |
| 6. A trapezoid ABCD has AB parallel to DC, and DC = AB + 100. The line joining the midpoints of AD and BC divides the trapezoid into two regions with areas in the ratio 2 : 3. Find the length of the segment parallel to DC that joins AD and BC and divides the trapezoid into two regions of equal area. |
| 7. Find 1/(2! 17!) + 1/(3! 16!) + ... + 1/(9! 10!). |
| 8. The trapezoid ABCD has AB parallel to DC, BC perpendicular to AB, and AC perpendicular to BD. Also AB = √11, AD = √1001. Find BC. |
| 9. z is a complex number such that z + 1/z = 2 cos 3o. Find [z2000 + 1/z2000] + 1. |
| 10. A circle radius r is inscribed in ABCD. It touches AB at P and CD at Q. AP = 19, PB = 26, CQ = 37, QD = 23. Find r. |
| 11. The trapezoid ABCD has AB and DC parallel, and AD = BC. A, D have coordinates (20,100), (21,107) respectively. No side is vertical or horizontal, and AD is not parallel to BC. B and C have integer coordinates. Find the possible slopes of AB. |
| 12. A, B, C lie on a sphere center O radius 20. AB = 13, BC = 14, CA = 15. `Find the distance of O from the triangle ABC. |
| 13. The equation 2000x6 + 100x5 + 10x3 + x - 2 = 0 has just two real roots. Find them. |
| 14. Every positive integer k has a unique factorial expansion k = a1 1! + a2 2! + ... + am m!, where m+1 > am > 0, and i+1 > ai ≥ 0. Given that 16! - 32! + 48! - 64! + ... + 1968! - 1984! + 2000! = a1 1! + a2 2! + ... + an n!, find a1 - a2 + a3 - a4 + ... + (-1)j+1 aj. |
| 15. Find the least positive integer n such that 1/(sin 45o sin 46o) + 1/(sin 47o sin 48o) + ... + 1/(sin 133o sin 134o) = 2/sin no. |
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
21 Aug 2003
Last updated/corrected 8 Dec 03