# 11 let n be the set 1 2 2018 for each subset a of n

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11. Let N be the set { 1 , 2 , . . . , 2018 } . For each subset A of N with exactly 1009 elements, define f ( A ) = X i A i X j N,j / A j. If E [ f ( A )] is the expected value of f ( A ) as A ranges over all the possible subsets of N with exactly 1009 elements, find the remainder when the sum of the distinct prime factors of E [ f ( A )] is divided by 1000. 12. Define a permutation of the set { 1 , 2 , 3 , ..., n } to be sortable if upon cancelling an appropriate term of such permutation, the remaining n - 1 terms are in increasing order. If f ( n ) is the number of sortable permutations of { 1 , 2 , 3 , ..., n } , find the remainder when 2018 X i =1 ( - 1) i · f ( i ) is divided by 1000. Note that the empty set is considered sortable. 13. Find the number of positive integers n < 2017 such that n 2 + n 0 + n 1 + n 7 is not divisible by the square of any prime. 14. Let 4 ABC be a triangle with AB = 6 , BC = 8 , AC = 10, and let D be a point such that if I A , I B , I C , I D are the incenters of the triangles BCD, ACD, ABD, ABC , respectively, the lines AI A , BI B , CI C , DI D are concurrent. If the volume of tetrahedron ABCD is 15 39 2 , then the sum of the distances from D to A, B, C can be expressed in the form a b for some positive relatively prime integers a, b . Find a + b . 15. A positive integer n is said to be m -free if n m ! and gcd( i, n ) = 1 for each i = 1 , 2 , ..., m . Let S k denote the sum of the squares of all the k -free integers. Find the remainder when S 7 - S 6 is divided by 1000.

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• Fall '19
• Prime number, Divisor

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