Euclidean geometry

USA AIME 1985 1 Let x1 = 97, and for n > 1 let xn = nxn?1 . Calculate the product x1x2 ? ? ?x8. 2 When a right triangle is rotated about one leg, the volume of the cone produced is 800pi cm3. When the triangle is rotated about the other leg, the volume of the cone produced is 1920pi cm3. What is the length (in cm) of the hypotenuse of the triangle? 3 Find c if a, b, and c are positive integers which satisfy c = (a+ bi)3 ? 107i, where i2 = ?1. 4 A small square is constructed inside a square of area 1 by dividing each side of the unit square into n equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of n if the the area of the small square is exactly 1/1985. A B CD 1/n