USA AIME 1987 1 An ordered pair (m,n) of non-negative integers is called ?simple? if the addition m+n in base 10 requires no carrying. Find the number of simple ordered pairs of non-negative integers that sum to 1492. 2 What is the largest possible distance between two points, one on the sphere of radius 19 with center (?2,?10, 5) and the other on the sphere of radius 87 with center (12, 8,?16)? 3 By a proper divisor of a natural number we mean a positive integral divisor other than 1 and the number itself. A natural number greater than 1 will be called ?nice? if it is equal to the product of its distinct proper divisors. What is the sum of the first ten nice numbers? 4 Find the area of the region enclosed by the graph of |x? 60|+ |y| = |x/4|.