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

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

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

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.
Solution

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!)
Solution

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