The division of segment Preliminary. Let some number be chosen, and one marks points {, {2, {3, {(n-1)on segment [0,1]. If the contrary isn' t expressly stated, we assume that is irrational. If = p/q, we assume that n<q. Thus, whatever is, points do not coincide. Thus, segment [0,1] is divided into n parts. Moreover let us assume that n>10 and 0.3<{}<0.7. This limitation is irrelevant, we impose it to exclude some trivial effects for small numbers. It means, in particular, that each part of division is less than {}. Note also that if you replace by n+or n-, then you get the same parts. So when the question concerns uniqueness, we suppose that 0<<1/2. The content of the task is to investigate what parts we get and how they are located. The tasks. The ratio of the lengths of the longest and the shortest segments, we denote by L = L( , n). A1. Let be the rational number, = p/q. Prove that there exist n for which L (n) = 1. A2. Does there exist some other integer or rational k, k>1, for which one may state that for any rational there exist n such that L (n) = k ? Now we reject the assumption that is rational. B1. Prove that for any n, there exist only 3 or less parts of different length. (Obviously, if is irrational, there are two or more different lengths).
Preliminary. Let some number be chosen, and one marks points {,
{2, {3, {(n-1)on segment [0,1]. If the contrary isn' t expressly
stated, we assume that is irrational. If = p/q, we assume that n10
and 0.310). Is it true that the assertions: (*) L takes the value A
only finite number of times and (**) L takes the value B only
finite number of times are equivalent?
