18 1 Methods of Proof
Now consider a sphere of the same radius as the planets. Remove from it all north
poles defined by directions that are perpendicular to the axes of two of the planets. This
is a set of area zero. For every other point on this sphere, there exists a direction in space
that makes it the north pole, and for that direction, there exists a unique north pole on
one of the planets that is not visible from the others. As such, the surface of the newly
introduced sphere is covered by patches translated from the other planets. Hence the total
area of invisible points is equal to the area of this sphere, which in turn is the area of one
of the planets.
58.Complete the square in Figure 6 with integers between 1 and 9 such that the sum
of the numbers in each row, column, and diagonal is as indicated.2
5
8
3(^16) 1321 25 27 20
30
26
16
14
Figure 6
59.Givennpoints in the plane, no three of which are collinear, show that there exists
a closed polygonal line with no self-intersections having these points as vertices.
60.Show that any polygon in the plane has a vertex, and a side not containing that
vertex, such that the projection of the vertex onto the side lies in the interior of the
side or at one of its endpoints.
61.In some country all roads between cities are one-way and such that once you leave
a city you cannot return to it again. Prove that there exists a city into which all
roads enter and a city from which all roads exit.
62.At a party assume that no boy dances with all the girls, but each girl dances with
at least one boy. Prove that there are two girl–boy couplesgbandg′b′who dance,
whereasbdoes not dance withg′, andgdoes not dance withb′.
63.The entries of a matrix are real numbers of absolute value less than or equal to 1,
and the sum of the elements in each column is 0. Prove that we can permute the
elements of each column in such a way that the sum of the elements in each row
will have absolute value less than or equal to 2.
64.Find all odd positive integersngreater than 1 such that for any coprime divisorsa
andbofn, the numbera+b−1 is also a divisor ofn.