14 1 Methods of Proof45.Each of nine straight lines divides a square into two quadrilaterals with the ratio of
their areas equal tor>0. Prove that at least three of these lines are concurrent.
46.Show that any convex polyhedron has two faces with the same number of edges.
47.Draw the diagonals of a 21-gon. Prove that at least one angle of less than 1◦is
formed.
48.LetP 1 ,P 2 ,...,P 2 nbe a permutation of the vertices of a regular polygon. Prove
that the closed polygonal lineP 1 P 2 ...P 2 ncontains a pair of parallel segments.
49.LetSbe a convex set in the plane that contains three noncollinear points. Each
point ofSis colored by one ofpcolors,p>1. Prove that for anyn≥3 there
exist infinitely many congruentn-gons whose vertices are all of the same color.
50.The points of the plane are colored by finitely many colors. Prove that one can find
a rectangle with vertices of the same color.
51.Inside the unit square lie several circles the sum of whose circumferences is equal
to 10. Prove that there exist infinitely many lines each of which intersects at least
four of the circles.1.4 Ordered Sets and Extremal Elements..............................
Anorderon a set is a relation≤with three properties: (i)a≤a; (ii) ifa≤bandb≤a,
thena=b; (iii)a≤bandb≤cimpliesa≤c. The order is called total if any two
elements are comparable, that is, if for everyaandb, eithera≤borb≤a. The simplest
example of a total order is≤on the set of real numbers. The existing order on a set can
be found useful when one is trying to solve a problem. This is the case with the following
two examples, the second of which is a problem of G. Galperin published in the Russian
journalQuantum.
Example.Prove that among any 50 distinct positive integers strictly less than 100 there
are two that are coprime.Solution.Order the numbers:x 1 <x 2 <···<x 50. If in this sequence there are two
consecutive integers, they are coprime and we are done. Otherwise,x 50 ≥x 1 + 2 · 49 =99.
Equality must hold, sincex 50 <100, and in this case the numbers are precisely the 50
odd integers less than 100. Among them 3 is coprime to 7. The problem is solved. Example.Given finitely many squares whose areas add up to 1, show that they can be
arranged without overlaps inside a square of area 2.