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Problem 11
11. Given any n>1 elements in the set {1, 2,..., 2n-1}, prove that one can always find two that are relatively prime.

Problem 12
12. Find an operation on the real numbers which is associative but not commutative.

Problem 13
13. Find x so that {frac(x.2^n); n in N} is dense in [0,1].

Problem 14
14. Given a positive integer n>1, show that it is always possible to find m > n/2 permutations f_1,...,f_m in S_n such that for all sigma in S_n, sigma^(-1)f_k has at least one fixed point for at least one index k. Show that it is impossible to find m <= n/2 such permutations.

Problem 15
15. Prove the multinomial formula: for all real numbers x_1,...,x_n and for all positive integer k, (x_1+...+x_n)^k=sum(C(|alpha|,alpha) x^alpha;|alpha|=k), where |alpha|=|(alpha_1,...,alpha_n)|=sum(|alpha_i|) (here the alpha_i are assumed to be nonnegative integers), x^alpha stands for x_1^alpha_1 ... x_n^alpha_n, and C(|alpha|,alpha) is the multinomial coefficient |alpha|!/(alpha_1! ... alpha_n!)

Problem 16
16. Find all positive integers a,b such that 4ab-a-b is a square.

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