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).
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
Let n be a positive integer exceeding 1. Determine
the number of permutations (a1, a2, ¼, an) of
(1, 2, ¼, n) for which there exists exactly one index
i with 1 £ i £ n and ai > ai+1.
Let (a1, a2, ¼, an) be an arithmetic
progression and (b1, b2, ¼, bn) be a geometric progression,
each of n positive real numbers, for which a1 = b1 and an = bn. Prove that
a1 + a2 + ¼+ an ³ b1 + b2 + ¼+ bn .|
For each positive integer k, let
ak = 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- ||
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
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.
Let f(x) be a convex realvalued function defined on
the reals, n ³ 2 and x1 < x2 < ¼ < xn. Prove that
x1 f(x2) + x2 f(x3) + ¼+ xn f(x1) ³ x2 f(x1) + x3 f(x2) + ¼+ x1 f(xn) .|