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Published online by Cambridge University Press: 31 January 2011
The Lion and the Christian. A lion and a Christian in a closed circular Roman arena have equal maximum speeds. What tactics should the lion employ to be sure of his meal? In other words, can the lion catch the Christian in finite time?
Integer Sequences
(i) Show that among n + 1 positive integers none of which is greater than 2n there are two such that one divides the other.
(ii) Show that among n + 1 positive integers none of which is greater than 2n there are two that are relatively prime.
(iii) Suppose that we have n natural numbers none of which is greater than 2n such that the least common multiple of any two is greater than 2n. Show that all n numbers are greater than 2n/3.
(iv) Show that every sequence of n = rs + 1 distinct integers with r, s ≥ 1 has an increasing subsequence of length r + 1 or a decreasing subsequence of length s + 1.
Points on a Circle
(i) Let X and Y be subsets of the vertex set of a regular n-gon. Show that there is a rotation ϱ of this polygon such that |X ∩ ϱ(Y)| ≥ |X||Y|/n, where, as usual, |Z| denotes the number of elements in a finite set Z.
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