Please send your solution to
Ms. Valeria Pendelieva
641 Kirkwood Avenue
Ottawa, ON K1Z 5X5
no later than December 15, 2003.
It is important that your complete mailing address
and your email address appear on the front page.

269.

Prove that the number
N = 2 ×4 ×6 ×¼×2000 ×2002+ 1 ×3 ×5 ×¼×1999 ×2001 

is divisible by 2003.

270.

A straight line cuts an acute triangle into two
parts (not necessarily triangles). In the same way, two other lines
cut each of these two parts into two parts. These steps repeat
until all the parts are triangles. Is it possible for all the
resulting triangle to be obtuse? (Provide reasoning to support
your answer.)

271.

Let x, y, z be natural numbers, such that the
number
is rational.
Prove that


(a) xz = y^{2};


(b) when y ¹ 1, the numbers x^{2} + y^{2} + z^{2}
and x^{2} + 4z^{2} are composite.

272.

Let ABCD be a parallelogram whose area is 2003 sq.
cm. Several points are chosen on the sides of the parallelogram.


(a) If there are 1000 points in addition to A, B, C, D,
prove that there always exist three points among these 1004
points that are vertices of a triangle whose area is less that
2 sq. cm.


(b) If there are 2000 points in addition to A, B, C, D, is
it true that there always exist three points among these 2004 points
that are vertices of a triangle whose area is less than 1 sq. cm?

273.

Solve the logarithmic inequality
log_{4} (9^{x}  3^{x}  1) ³ log_{2} Ö5 . 


274.

The inscribed circle of an isosceles triangle
ABC is tangent to the side AB at the point T and
bisects the segment CT. If CT = 6Ö2, find the
sides of the triangle.

275.

Find all solutions of the trigonometric equation
sinx  sin3x + sin5x = cosx  cos3x + cos5x . 
