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# 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 $x$, $⌊x⌋$ (the floor of $x$) is equal to the greatest integer that is less than or equal to $x$.

25.
Let $a$, $b$, $c$ be non-negative numbers such that $a+b+c=1$. Prove that

$\frac{\mathrm{ab}}{c+1}+\frac{\mathrm{bc}}{a+1}+\frac{\mathrm{ca}}{b+1}\le \frac{1}{4} .$

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 $‖{36}^{m}-{5}^{n}‖$ 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 $‖f\left(x\right)-f\left(y\right)‖<‖x-y‖$ for every $x,y\in A$, $x\ne y$. Prove that there is $a\in A$ for which $f\left(a\right)=a$. 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 $⌊\sqrt{a}⌋$ 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.