Advanced book on Mathematics Olympiad

2.4 Abstract Algebra 93

a∗b=a(b^2 b).

Prove that(M,∗)is a group.

284.Givena finite multiplicative group of matrices with complex entries, denote by
Mthe sum of the matrices in. Prove that detMis an integer.
We would like to point out the following property of the set of real numbers.

Theorem.A nontrivial subgroup of the additive group of real numbers is either cyclic
or it is dense in the set of real numbers.

Proof.Denote the group byG. It is either discrete, or it has an accumulation point on the
real axis. If it is discrete, letabe its smallest positive element. Then any other element
is of the formb=ka+αwith 0≤α<a. Butbandkaare both inG; henceαis inG
as well. By the minimality ofa,αcan only be equal to 0, and hence the group is cyclic.
If there is a sequence(xn)ninGconverging to some real number, then±(xn−xm)
approaches zero asn, m→∞. Choosing the indicesmandnappropriately, we can find
a sequence of positive elements inGthat converges to 0. Thus for any>0 there is an
elementc∈Gwith 0<c<. For some integerk, the distance betweenkcand(k+ 1 )c
is less than; hence any interval of lengthcontains some multiple ofc. Varying,we
conclude thatGis dense in the real axis. 

Try to use this result to solve the following problems.

285.Letf :R → Rbe a continuous function satisfyingf(x)= f(x+

√

2 ) =

f(x+

√

3 )for allx. Prove thatfis constant.

286.Prove that the sequence(sinn)nis dense in the interval[− 1 , 1 ].

287.Show that infinitely many powers of 2 start with the digit 7.

288.Given a rectangle, we are allowed to fold it in two or in three, parallel to one side
or the other, in order to form a smaller rectangle. Prove that for any>0 there are
finitely many such operations that produce a rectangle with the ratio of the sides
lying in the interval( 1 −, 1 +)(which means that we can get arbitrarily close
to a square).

289.A set of points in the plane is invariant under the reflections across the sides of some
given regular pentagon. Prove that the set is dense in the plane.

“There is no certainty in sciences where one of the mathematical sciences cannot be
applied, or which are not in relation with this mathematics.’’ This thought of Leonardo
da Vinci motivated us to include an example of how groups show up in natural sciences.
The groups of symmetries of three-dimensional space play an important role in chem-
istry and crystallography. In chemistry, the symmetries of molecules give rise to physical