location:

# 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 ${90}^{ˆ}$. 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

$\frac{1}{\sqrt{{x}_{1}}}+\frac{1}{\sqrt{{x}_{2}}}+\dots +\frac{1}{\sqrt{{x}_{100}}}=20 .$

Prove that at least two of the numbers are equal.

22.
Let R be a rectangle with dimensions $11×12$. Find the least natural number $n$ for which it is possible to cover R with $n$ rectangles, each of size $1×6$ or $1×7$, 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 ${120}^{ˆ}$ 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.