PROBLEMS FOR JUNE
Please send your solution to
Ms. Valeria Pendelieva
641 Kirkwood Avenue
Ottawa, ON K1Z 5X5
no later than August 10, 2003.
It is important that your complete mailing address
and your email address appear on the front page.

234.

A square of side length 100 is divided into
10000 smaller unit squares. Two squares sharing a common
side are called neighbours.


(a) Is it possible to colour an even number of squares so that
each coloured square has an even number of coloured neighbours?


(b) Is it possible to colour an odd number of squares so that
each coloured square has an odd number of coloured neighbours?

235.

Find all positive integers, N, for which:


(i) N has exactly sixteen positive divisors:
1 = d_{1} < d_{2} < ¼ < d_{16} = N;


(ii) the divisor with the index d_{5} (namely,
d_{d5}) is equal to (d_{2} + d_{4})×d_{6} (the product of the
two).

236.

For any positive real numbers a, b, c, prove that

1
b(a + b)

+ 
1
c(b + c)

+ 
1
a(c + a)

³ 
27
2(a + b + c)^{2}

. 


237.

The sequence { a_{n} : n = 1, 2, ¼} is defined
by the recursion
a_{n+2} = 3a_{n+1}  a_{n} for n ³ 1 . 

Find all natural numbers n for which 1 + 5a_{n} a_{n+1} is a
perfect square.

238.

Let ABC be an acuteangled triangle, and let M be
a point on the side AC and N a point on the side BC. The
circumcircles of triangles CAN and BCM intersect at the two
points C and D. Prove that the line CD passes through the
circumcentre of triangle ABC if and only if the right bisector of
AB passes through the midpoint of MN.

239.

Find all natural numbers n for which the
diophantine equation
has positive integer solutions x, y, z.

240.

In a competition, 8 judges rate each contestant
"yes" or "no". After the competition, it turned out, that
for any two contestants, two judges marked the first one by
"yes" and the second one also by "yes"; two judges have
marked the first one by "yes" and the second one by "no";
two judges have marked the first one by "no" and the second
one by "yes"; and, finally, two judges have marked the first
one by "no" and the second one by "no". What is the
greatest number of contestants?