PROBLEMS FOR JANUARY
Please send your solution to
Ed Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
no later than February 29, 2004.
It is important that your complete mailing address
and your email address appear on the front page.

283.

(a) Determine all quadruples (a, b, c, d)
of positive integers for which the greatest common divisor
of its elements is 1,
and a + b + c = d.


(b) Of those quadruples found in (a), which also satisfy


(c) For quadruples (a, b, c, d) of positive integers,
do the conditions a + b + c = d and (1/b) + (1/c) + (1/d) = (1/a)
together imply that a/b = c/d?

284.

Suppose that ABCDEF is a convex hexagon for which
ÐA + ÐC + ÐE = 360^{°} and

AB
BC

· 
CD
DE

· 
EF
FA

= 1 . 

Prove that

AB
BF

· 
FD
DE

· 
EC
CA

= 1 . 


285.

(a) Solve the following system of equations:
(1 + 4^{2x  y})(5^{1  2x + y}) = 1 + 2^{2x  y + 1} ; 

y^{2} + 4x = log_{2} (y^{2} + 2x + 1) . 



(b) Solve for real values of x:
Express your answers in a simple form.

286.

Construct inside a triangle ABC a point P such that,
if X, Y, Z are the respective feet of the perpendiculars from
P to BC, CA, AB, then P is the centroid (intersection of
the medians) of triangle XYZ.

287.

Let M and N be the respective midpoints of the
sides BC and AC of the triangle ABC. Prove that the centroid
of the triangle ABC lies on the circumscribed circle of the triangle
CMN if and only if
4 ·AM ·BN  = 3 ·AC ·BC  . 


288.

Suppose that a_{1} < a_{2} < ¼ < a_{n}. Prove that
a_{1} a_{2}^{4} + a_{2} a_{3}^{4} + ¼+ a_{n} a_{1}^{4} ³ a_{2} a_{1}^{4} + a_{3} a_{2}^{4} + ¼+ a_{1} a_{n}^{4} . 


289.

Let n(r) be the number of points with integer
coordinates on the circumference of a circle of radius r > 1
in the cartesian plane. Prove that