1. Ex1:
Points with integer coordinates in cartesian space are called lattice points. We color
2000 lattice points blue and 2000 other lattice points red in such a way that no two
blue-red segments have a common interior point (a segment is blue-red if its two
endpoints are colored blue and red). Consider the smallest rectangular
parallelepiped that covers all the colored points.
(a) Prove that this rectangular parallelepiped covers at least 500,000 lattice points.
(b) Give an example of a coloring for which the considered rectangular
paralellepiped covers at most 8,000,000lattice points.
Ex2:
Arrange arbitrarily 1,2,,25 on a circumference. We consider all 25 sums obtained
by adding 5 consecutive numbers. If the number of distinct residues of those sums
modulo 5 is d (0  d  5) ,find all possible values of d.
Ex3:
Two persons, A and B, set up an incantation contest in which they spell
incantations (i.e. a finite sequence of letters) alternately. They must obey the
following rules:
i) Any incantation can appear no more than once;
ii) Except for the first incantation, any incantation must be obtained by permuting
the letters of the last one before it, or deleting one letter from the last incantation
before it;
iii)The first person who cannot spell an incantation loses the contest. Answer the
following questions:
a) If A says ' STAGEPREIMO' first, then who will win?
b) Let M be the set of all possible incantations whose lengths (i.e. the numbers of
letters in them) are 2009 and containing only four letters A,B,C,D , each of them
appearing at least once. Find the first incantation (arranged in dictionary order) in
M such that A has a winning strategy by starting with it.