| 1. The sequence a1, a2, ... , a98 satisfies an+1 = an + 1 for n = 1, 2, ... , 97 and has sum 137. Find a2 + a4 + a6 + ... + a98. |
| 2. Find the smallest positive integer n such that every digit of 15n is 0 or 8. |
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3. P is a point inside the triangle ABC. Lines are drawn through P parallel to the sides of the triangle. The areas of the three resulting triangles with a vertex at P have areas 4, 9 and 49. What is the area of ABC?
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| 4. A sequence of positive integers includes the number 68 and has arithmetic mean 56. When 68 is removed the arithmetic mean of the remaining numbers is 55. What is the largest number than can occur in the sequence? |
| 5. The reals x and y satisfy log8x + log4(y2) = 5 and log8y + log4(x2) = 7. Find xy. |
| 6. Three circles radius 3 have centers at P (14, 92), Q (17, 76) and R (19, 84). The line L passes through Q and the total area of the parts of the circles in each half-plane (defined by L) is the same. What is the absolute value of the slope of L? |
| 7. Let Z be the integers. The function f : Z → Z satisfies f(n) = n - 3 for n > 999 and f(n) = f( f(n+5) ) for n < 1000. Find f(84). |
| 8. z6 + z3 + 1 = 0 has a root r eiθ with 90o < θ < 180o. Find θ. |
| 9. The tetrahedron ABCD has AB = 3, area ABC = 15, area ABD = 12 and the angle between the faces ABC and ABD is 30o. Find its volume. |
| 10. An exam has 30 multiple-choice problems. A contestant who answers m questions correctly and n incorrectly (and does not answer 30 - m - n questions) gets a score of 30 + 4m - n. A contestant scores N > 80. A knowledge of N is sufficient to deduce how many questions the contestant scored correctly. That is not true for any score M satisfying 80 < M < N. Find N. |
| 11. Three red counters, four green counters and five blue counters are placed in a row in random order. Find the probability that no two blue counters are adjacent. |
| 12. Let R be the reals. The function f : R → R satisfies f(0) = 0 and f(2 + x) = f(2 - x) and f(7 + x) = f(7 - x) for all x. What is the smallest possible number of values x such that |x| ≤ 1000 and f(x) = 0? |
| 13. Find 10 cot( cot-13 + cot-17 + cot-113 + cot-121). |
| 14. What is the largest even integer that cannot be written as the sum of two odd composite positive integers? |
| 15. The real numbers x, y, z, w satisfy: x2/(n2 - 12) + y2/(n2 - 32) + z2/(n2 - 52) + w2/(n2 - 72) = 1 for n = 2, 4, 6 and 8. Find x2 + y2 + z2 + w2. |
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
19 March 2003
Last updated/corrected 1 Aug 03