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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 $11 \times 12$ . 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 $120^{ˆ}$ at the centre.

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