USA AIME 1983 1 Let x, y, and z all exceed 1 and let w be a positive number such that logxw = 24, logy w = 40, and logxyz w = 12. Find logz w. 2 Let f(x) = |x? p|+ |x? 15|+ |x? p? 15|, where 0 < p < 15. Determine the minimum value taken by f(x) for x in the interval p ? x ? 15. 3 What is the product of the real roots of the equation x2 + 18x+ 30 = 2 ? x2 + 18x+ 45? 4 A machine-shop cutting tool has the shape of a notched circle, as shown. The radius of the circle is ? 50 cm, the length of AB is 6 cm, and that of BC is 2 cm. The angle ABC is a right angle. Find the square of the distance (in centimeters) from B to the center of the circle. A B C 5 Suppose that the sum of the squares of two complex numbers x and y is 7 and the sum of