PROBLEMS FOR AUGUST
Solutions should be submitted to
Dr. Dragos Hrimiuc
Department of Mathematics
University of Alberta
Edmonton, AB T6G 2G1
Solution to these problems should be
postmarked no later than September 30, 2000.
Note: For any real number
$x$,
$\lfloor x\rfloor $
(the floor of
$x$) is equal to the greatest integer
that is less than or equal to
$x$.

25.

Let
$a$,
$b$,
$c$ be nonnegative numbers such that
$a+b+c=1$. Prove that
$\frac{\mathrm{ab}}{c+1}+\frac{\mathrm{bc}}{a+1}+\frac{\mathrm{ca}}{b+1}\le \frac{1}{4}\hspace{1em}\hspace{1em}.$
When does equality hold?

26.

Each of
$m$ cards is labelled by one of the numbers
$1,2,\dots ,m$. Prove that, if the sum of labels of any
subset of cards is not a multiple of
$m+1$, then each card is
labelled by the same number.
27.

Find the least number of the form
$\Vert {36}^{m}{5}^{n}\Vert $
where
$m$ and
$n$ are positive integers.

28.

Let
$A$ be a finite set of real numbers which contains at
least two elements and let
$f:A\to A$ be a function such
that
$\Vert f(x)f(y)\Vert <\Vert xy\Vert $ for every
$x,y\in A$,
$x\ne y$. Prove that there is
$a\in A$ for which
$f(a)=a$. Does the result remain valid if
$A$ is not a finite set?

29.

Let
$A$ be a nonempty set of positive integers such that
if
$a\in A$, then
$4a$ and
$\lfloor \sqrt{a}\rfloor $
both belong to
$A$. Prove that
$A$ is the set of all positive integers.

30.

Find a point
$M$ within a regular pentagon for which the sum of
its distances to the vertices is minimum.