Subcontests
(4)rearrangement of sums, is it possible?
For which positive integers n is the following statement true:
if a1,a2,...,an are positive integers, ak≤n for all k and k=1∑nak=2n
then it is always possible to choose ai1,ai2,...,aij in such a way that
the indices i1,i2,...,ij are different numbers, and k=1∑jaik=n? x_0 = t, x_{n+1} = 2x_n^2-1, x_n = 1
Let S be the set of all points t in the closed interval [−1,1] such that for the sequence x0,x1,x2,... defined by the equations x0=t,xn+1=2xn2−1, there exists a positive integer N such that xn=1 for all n≥N. Show that the set S has infinitely many elements.