456 Algebra
Canceling the powers of 2, this amounts to showing that{^2
n
5 m|m, nintegers}is dense.
We further simplify the problem by applying the function log 2 to the fraction. This
function is continuous, so it suffices to prove that{n−mlog 25 |m, nintegers}is dense
on the real axis. This is an additive group, which is not cyclic since log 2 5 is irrational
(and so 1 and log 2 5 cannot both be integer multiples of the same number). It follows
that this group is dense in the real numbers, and the problem is solved.
(V.I. Arnol’d,Mathematical Methods of Classical Mechanics, Springer-Verlag, 1997)
288.Ifris the original ratio of the sides, after a number of folds the ratio will be 2m 3 nr,
wheremandnare integer numbers. It suffices to show that the set{ 2 m 3 nr|m, n∈Z}
is dense in the positive real axis. This is the same as showing that{ 2 m 3 n|m, n∈Z}is
dense. Taking the logarithm, we reduce the problem to the fact that the additive group
{m+nlog 23 |m, n∈Z}is dense in the real axis. And this is true since the group is not
cyclic.
(German Mathematical Olympiad)
289.Call the regular pentagonABCDEand the set. Composing a reflection across
ABwith a reflection acrossBC, we can obtain a 108◦rotation aroundB. The set
is invariant under this rotation. There is a similar rotation aroundC, of the same angle
and opposite direction, which also preserves. Their composition is a translation by
a vector that makes an angle of 36◦withBCand has length 2 sin 54◦BC. Figure 65
helps us understand why this is so. Indeed, ifProtates toP′aroundB, andP′toP′′
aroundC, then the triangleP′BCtransforms to the triangleP′PP′′by a rotation around
P′of angle∠CP′P′′ = 36 ◦followed by a dilation of ratioP′P′′/P′C =2 sin 54◦.
Note that the translation preserves the set. Reasoning similarly with verticesAand
BC
P
P
P"
Figure 65D, and taking into account thatADis parallel toBC, we find a translation by a vector
of length 2 sin 54◦ADthat makes an angle of 36◦withBCand preserves. Because
AD/BC=2 sin 54◦=
√ 5 + 1
2 , the groupGBCgenerated by the two translations is dense
in the group of all translations by vectors that make an angle of 36◦withBC. The same