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

s(n)

s(k)

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 M₃, then there exist integers a Î M₁ and b Î M₂ 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

tan² 2x = 2 tan2x tan3x + 1 .