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Type system

AHSME 1988

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Algebra
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Mathematics
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Type system

USA AIME 1988 1 One commercially available ten-button lock may be opened by depressing ? in any order ? the correct five buttons. The sample shown below has {1, 2, 3, 6, 9} as its combination. Suppose that these locks are redesigned so that sets of as many as nine buttons or as few as one button could serve as combinations. How many additional combinations would this allow? 5 10 4 9 3 8 2 7 1 6 2 For any positive integer k, let f1(k) denote the square of the sum of the digits of k. For n ? 2, let fn(k) = f1(fn?1(k)). Find f1988(11). 3 Find (log2 x)2 if log2(log8 x) = log8(log2 x). 4 Suppose that |xi| < 1 for i = 1, 2, . . . , n. Suppose further that |x1|+ |x2|+ ? ? ?+ |xn| = 19 + |x1 + x2 + ? ? ?+ xn|. What is the smallest possible value of n?
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