location:

 PROBLEMS FOR JANUARY

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

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, an) of (1, 2, ¼, n) for which there exists exactly one index i with 1 £ i £ n and ai > ai+1.

201.
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 .

202.
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- 12 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 < xn. Prove that

 x1 f(x2) + x2 f(x3) + ¼+ xn f(x1) ³ x2 f(x1) + x3 f(x2) + ¼+ x1 f(xn) .
 top of page | contact us | privacy | site map |