PROBLEMS FOR JULY
Solutions should be submitted to
Dr. Valeria Pandelieva
708  195 Clearview Avenue
Ottawa, ON K1Z 6S1
Solution to these problems should be
postmarked no later than August 30, 2000.
Notes: An acute triangle has all of its
angles less than
${90}^{\u02c6}$. The orthocentre
of a triangle is the intersection point of its altitudes.
Points are collinear iff they lie on a straight line.

19.

Is it possible to divide the natural numbers
$1,2,\dots ,n$ into two groups, such that the squares
of the members in each group have the same sum, if
(a)
$n=40000$; (b)
$n=40002$? Explain your answer.

20.

Given any six irrational numbers, prove that
there are always three of them, say
$a$,
$b$,
$c$, for which
$a+b$,
$b+c$ and
$c+a$ are irrational.

21.

The natural numbers
${x}_{1}$,
${x}_{2}$,
$\dots $,
${x}_{100}$
are such that
$\frac{1}{\sqrt{{x}_{1}}}+\frac{1}{\sqrt{{x}_{2}}}+\dots +\frac{1}{\sqrt{{x}_{100}}}=20\hspace{1em}.$
Prove that at least two of the numbers are equal.

22.

Let R be a rectangle with dimensions
$11\times 12$.
Find the least natural number
$n$ for which it is possible to cover
R with
$n$ rectangles, each of size
$1\times 6$ or
$1\times 7$,
with no two of these having a common interior point.

23.

Given 21 points on the circumference of a circle,
prove that at least 100 of the arcs determined by pairs of
these points subtend an angle not exceeding
${120}^{\u02c6}$ at
the centre.

24.

$\mathrm{ABC}$ is an acute triangle with orthocentre
$H$.
Denote by
$M$ and
$N$ the midpoints of the respective segments
$\mathrm{AB}$ and
$\mathrm{CH}$, and by
$P$ the intersection point of the
bisectors of angles
$\mathrm{CAH}$ and
$\mathrm{CBH}$. Prove that the points
$M$,
$N$ and
$P$ are collinear.