**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

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