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 30, 2003. It is important that your complete mailing address and your email address appear on the front page.

213.

Suppose that each side and each diagonal of a regular hexagon A₁A₂A₃A₄A₅A₆ is coloured either red or blue, and that no triangle A_iA_jA_k has all of its sides coloured blue. For each k = 1, 2, ¼, 6, let r_k be the number of segments A_kA_j (j ¹ k) coloured red. Prove that

6
å
k=1

(2r_k - 7)² £ 54 .

214.: Let S be a circle with centre O and radius 1, and let P_i (1 £ i £ n) be points chosen on the (circumference of the) circle for which å_i=1ⁿ[( ®) || ( OP_i)] = 0. Prove that, for each point X in the plane, å|XP_i | ³ n.

215.: Find all values of the parameter a for which the equation 16x⁴ - ax³ + (2a + 17)x² - ax + 16 = 0 has exactly four real solutions which are in geometric progression.

216.

Let x be positive and let 0 < a £ 1. Prove that

(1 - x^a)(1 - x)^-1 £ (1 + x )^a-1.

217.: Let the three side lengths of a scalene triangle be given. There are two possible ways of orienting the triangle with these side lengths, one obtainable from the other by turning the triangle over, or by reflecting in a mirror. Prove that it is possible to slice the triangle in one of its orientations into finitely many pieces that can be rearranged using rotations and translations in the plane (but not reflections and rotations out of the plane) to form the other.

218.: Let ABC be a triangle. Suppose that D is a point on BA produced and E a point on the side BC, and that DE intersects the side AC at F. Let BE + EF = BA + AF. Prove that BC + CF = BD + DF.

219.: There are two definitions of an ellipse.

(1) An ellipse is the locus of points P such that the sum of its distances from two fixed points F₁ and F₂ (called foci) is constant.

(2) An ellipse is the locus of points P such that, for some real number e (called the eccentricity) with 0 < e < 1, the distance from P to a fixed point F (called a focus) is equal to e times its perpendicular distance to a fixed straight line (called the directrix).

Prove that the two definitions are compatible.