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Ring theory

AHSME 1989

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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.
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