PROBLEMS FOR MARCH
Please send your solution to
Edward J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
no later than April 25, 2004.
It is important that your complete mailing address
and your email address appear on the front page.

297.

The point P lies on the side BC of triangle
ABC so that PC = 2BP, ÐABC = 45^{°} and ÐAPC = 60^{°}. Determine ÐACB.

298.

Let O be a point in the interior of a quadrilateral
of area S, and suppose that
2S = OA ^{2} + OB ^{2} + OC ^{2}+ OD ^{2} . 

Prove that ABCD is a square with centre O.

299.

Let s(r) denote the sum of all the divisors
of r, including r and 1. Prove that there are infinitely
many natural numbers n for which
whenever 1 £ k £ n.

300.

Suppose that ABC is a right triangle with
ÐB < ÐC < ÐA = 90^{°}, and let
\frak K be its circumcircle. Suppose that the tangent
to \frak K at A meets BC produced at D and that E is
the reflection of A in the axis BC. Let X be the foot
of the perpendicular for A to BE and Y the midpoint of AX.
Suppose that BY meets \frak K again in Z. Prove that
BD is tangent to the circumcircle of triangle ADZ.

301.

Let d = 1, 2, 3. Suppose that M_{d} consists of the
positive integers that cannot be expressed as the sum of
two or more consecutive terms of an arithmetic progression
consisting of positive integers with common difference d.
Prove that, if c Î M_{3}, then there exist integers a Î M_{1} and b Î M_{2} for which c = ab.

302.

In the following, ABCD is an arbitrary convex
quadrilateral. The notation [ ¼] refers to the area.


(a) Prove that ABCD is a trapezoid if and only if
[ABC] ·[ACD] = [ABD] ·[BCD] . 



(b) Suppose that F is an interior point of the
quadrilateral ABCD such that ABCF is a parallelogram.
Prove that
[ABC] ·[ACD] + [AFD] ·[FCD] = [ABD] ·[BCD] . 


303.

Solve the equation
tan^{2} 2x = 2 tan2x tan3x + 1 . 
