Problem:
The length of the edge of the cube is 1. Two points M and N move along the line segments AB and
, respectively, in such a way that at any time t
we have
and
. Show that MN has no maximum.
Proof:
Clearly, . If
for some t, then
and
, which is equivalent to
and
. Consequently,
and
for some integers k and n, which implies
, a contradiction since
is irrational. This is why
for any t.
We will now show that MN can be made arbitrarily close to 1. For any integer k, set . Then,
, so at any time
, the point M is at A. To show that N can be arbitrarily close to
at times
, it is enough to show that
can be arbitrarily close to 1 for appropriate choices of k.
We are now going to use Kronecker’s theorem:
If is an irrational number, then the set of numbers of the form
, where m is a positive integer while n is an arbitrary integer, is dense in the set of all real numbers. The latter means that every non-empty open interval (regardless of how small it is) contains a number of the form
.
Since is irrational, we can use Kronecker’s theorem with
. Then, for
, and any
, there exist integer
and
such that
. That is, for
we have
. Since
, we have
.
It remains to note that tends to 1 as
tends to 0.
Hence, MN can be made arbitrarily close to 1.
Ref: Geometric Problems on Maxima and Minima by Titu Andreescu, Oleg Mushkarov, and Luchezar Stoyanov.
Thanks to Prof Andreescu, et al !
Nalin Pithwa