Area

USA AIME 1989 1 Compute ? (31)(30)(29)(28) + 1. 2 Ten points are marked on a circle. How many distinct convex polygons of three or more sides can be drawn using some (or all) of the ten points as vertices? 3 Suppose n is a positive integer and d is a single digit in base 10. Find n if n 810 = 0.d25d25d25 . . . 4 If a < b < c < d < e are consecutive positive integers such that b + c + d is a perfect square and a+ b+ c+ d+ e is a perfect cube, what is the smallest possible value of c? 5 When a certain biased coin is flipped five times, the probability of getting heads exactly once is not equal to 0 and is the same as that of getting heads exactly twice. Let ij , in lowest terms, be the probability that the coin comes up heads in exactly 3 out of 5 flips. Find i+ j.

USA AIME 1984 1 Find the value of a2 + a4 + a6 + ? ? ?+ a98 if a1, a2, a3, . . . is an arithmetic progression with common difference 1, and a1 + a2 + a3 + ? ? ?+ a98 = 137. 2 The integer n is the smallest positive multiple of 15 such that every digit of n is either 8 or 0. Compute n15 . 3 A point P is chosen in the interior of 4ABC so that when lines are drawn through P parallel to the sides of 4ABC, the resulting smaller triangles, t1, t2, and t3 in the figure, have areas 4, 9, and 49, respectively. Find the area of 4ABC. A B C t3 t2t1 4 Let S be a list of positive integers - not necessarily distinct - in which the number 68 appears. The average (arithmetic mean) of the numbers in S is 56. However, if 68 is removed, the