| 1. x, y, z are real numbers greater than 1 and w is a positive real number. If logxw = 24, logyw = 40 and logxyzw = 12, find logzw. |
| 2. Find the minimum value of |x - p| + |x - 15| + |x - p - 15| for x in the range p ≤ x ≤ 15, where 0 < p < 15. |
| 3. Find the product of the real roots of the equation x2 + 18x + 30 = 2 √(x2 + 18x + 45). |
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4. A and C lie on a circle center O with radius √50. The point B inside the circle is such that ∠ABC = 90o, AB = 6, BC = 2. Find OB.
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| 5. w and z are complex numbers such that w2 + z2 = 7, w3 + z3 = 10. What is the largest possible real value of w + z? |
| 6. What is the remainder on dividing 683 + 883 by 49? |
| 7. 25 knights are seated at a round table and 3 are chosen at random. Find the probability that at least two of the chosen 3 are sitting next to each other. |
| 8. What is the largest 2-digit prime factor of the binomial coefficient 200C100? |
| 9. Find the minimum value of (9x2sin2x + 4)/(x sin x) for 0 < x < π. |
| 10. How many 4 digit numbers with first digit 1 have exactly two identical digits (like 1447, 1005 or 1231)? |
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11. ABCD is a square side 6√2. EF is parallel to the square and has length 12√2. The faces BCF and ADE are equilateral. What is the volume of the solid ABCDEF?
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12. The chord CD is perpendicular to the diameter AB and meets it at H. The distances AB and CD are integral. The distance AB has 2 digits and the distance CD is obtained by reversing the digits of AB. The distance OH is a non-zero rational. Find AB.
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| 13. For each non-empty subset of {1, 2, 3, 4, 5, 6, 7} arrange the members in decreasing order with alternating signs and take the sum. For example, for the subset {5} we get 5. For {6, 3, 1} we get 6 - 3 + 1 = 4. Find the sum of all the resulting numbers. |
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14. The distance AB is 12. The circle center A radius 8 and the circle center B radius 6 meet at P (and another point). A line through P meets the circles again at Q and R (with Q on the larger circle), so that QP = PR. Find QP2.
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15. BC is a chord length 6 of a circle center O radius 5. A is a point on the circle closer to B than C such that there is just one chord AD which is bisected by BC. Find sin AOB.
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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
14 March 2003
Last updated/corrected 26 Nov 03