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## PROBLEMS FOR JANUARY

Ed Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3

283.
(a) Determine all quadruples (a, b, c, d) of positive integers for which the greatest common divisor of its elements is 1,
 a b = c d
and a + b + c = d.
(b) Of those quadruples found in (a), which also satisfy
 1 b + 1 c + 1 d = 1 a ?
(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 + 42x - y)(51 - 2x + y) = 1 + 22x - y + 1 ;

 y2 + 4x = log2 (y2 + 2x + 1) .
(b) Solve for real values of x:
 3x ·8x/(x+2) = 6 .

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 a1 < a2 < ¼ < an. Prove that
 a1 a24 + a2 a34 + ¼+ an a14 ³ a2 a14 + a3 a24 + ¼+ a1 an4 .

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
 n(r) < 6 3 Ö pr2 .

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