CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  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. 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, , 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 x1, x2, , x100 are such that
1
  __
Íx1
+ 1
  __
Íx2
+ + 1

Í

x100
= 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 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.


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