Please send your solution to

Edward J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3

no later than February 21, 2003. It is important that your complete mailing address and your email address appear on the front page.

Notes. A function is convex if and only if for each u and v, and for each t Î [0, 1], f(tu + (1-t)v) £ tf(u) + (1-t)f(v).

199.

Let A and B be two points on a parabola with vertex V such that VA is perpendicular to VB and q is the angle between the chord VA and the axis of the parabola. Prove that

|VA | |VB | = cot3 q .

200.

Let n be a positive integer exceeding 1. Determine the number of permutations (a₁, a₂, ¼, a_n) of (1, 2, ¼, n) for which there exists exactly one index i with 1 £ i £ n and a_i > a_i+1.

201.

Let (a₁, a₂, ¼, a_n) be an arithmetic progression and (b₁, b₂, ¼, b_n) be a geometric progression, each of n positive real numbers, for which a₁ = b₁ and a_n = b_n. Prove that

a1 + a2 + ¼+ an ³ b1 + b2 + ¼+ bn .

202.

For each positive integer k, let a_k = 1 + (1/2) + (1/3) + ¼+ (1/k). Prove that, for each positive integer n,

3a1 + 5a2 + 7a3 + ¼+ (2n + 1)an = (n + 1)2 an- 1 2 n(n+1) .

203.

Every midpoint of an edge of a tetrahedron is contained in a plane that is perpendicular to the opposite edge. Prove that these six planes intersect in a point that is symmetric to the centre of the circumsphere of the tetrahedron with respect to its centroid.

204.

Each of n ³ 2 people in a certain village has at least one of eight different names. No two people have exactly the same set of names. For an arbitrary set of k names, where 1 £ k £ 7, the number of people containing at least one of the k names among his/her set of names is even. Determine the value of n.

205.

Let f(x) be a convex realvalued function defined on the reals, n ³ 2 and x₁ < x₂ < ¼ < x_n. Prove that

x1 f(x2) + x2 f(x3) + ¼+ xn f(x1) ³ x2 f(x1) + x3 f(x2) + ¼+ x1 f(xn) .