PROBLEMS FOR MAY
Please send your solution to
Ms. Valeria Pendelieva
641 Kirkwood Avenue
Ottawa, ON K1Z 5X5
no later than June 30, 2004.
It is important that your complete mailing address
and your email address appear on the front page.
Notes. The absolute value z of the number z is defined by

311.

Given a square with a side length 1, let P
be a point in the plane such that the sum of the distances from P
to the sides of the square (or their extensions) is equal to 4. Determine the set of
all such points P.

312.

Given ten arbitrary natural numbers. Consider the
sum, the product, and the absolute value of the difference
calculated for any two of these numbers. At most how many of
all these calculated numbers are odd?

313.

The three medians of the triangle ABC partition it
into six triangles. Given that three of these triangles have
equal perimeters, prove that the triangle ABC is equilateral.

314.

For the real numbers a, b and c, it is known that
and
Find the value of the expression
M = 
1
1 + a + ab

+ 
1
1 + b + bc

+ 
1
1 + c + ca

. 


315.

The natural numbers 3945, 4686 and 5598 have the
same remainder when divided by a natural number x. What is the
sum of the number x and this remainder?

316.

Solve the equation
x^{2}  3x + 2 + x^{2} + 2x  3  = 11 . 


317.

Let P(x) be the polynomial
P(x) = x^{15}  2004x^{14} + 2004x^{13}  ¼ 2004x^{2} + 2004x , 

Calculate P(2003).