PROBLEMS FOR FEBRUARY
Please send your solution to
Edward J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
no later than March 31, 2003.
It is important that your complete mailing address
and your email address appear on the front page.

206.

In a group consisting of five people, among
any three people, there are two who know each other and two
neither of whom knows the other. Prove that it is possible
to seat the group around a circular table so that each
adjacent pair knows each other.

207.

Let n be a positive integer exceeding 1.
Suppose that A = (a_{1}, a_{2}, ¼, a_{m}) is an ordered set
of m = 2^{n} numbers, each of which is equal to either 1 or
1. Let
S(A) = (a_{1} a_{2}, a_{2} a_{3}, ¼, a_{m1} a_{m}, a_{m} a_{1}) . 

Define, S^{0} (A) = A, S^{1} (A) = S(A), and for k ³ 1,
S^{k+1} = S(S^{k} (A)). Is it always possible to find a positive
integer r for which S^{r} (A) consists entirely of 1s?

208.

Determine all positive integers n for which
n = a^{2} + b^{2} + c^{2} + d^{2}, where a < b < c < d and
a, b, c, d are the four smallest positive divisors of n.

209.

Determine all positive integers n for which
2^{n}  1 is a multiple of 3 and (2^{n}  1)/3 has a multiple
of the form 4m^{2} + 1 for some integer m.

210.

ABC and DAC are two isosceles triangles for which
B and D are on opposite sides of AC, AB = AC, DA = DC
ÐBAC = 20^{°} and ÐADC = 100^{°}.
Prove that AB = BC + CD.

211.

Let ABC be a triangle and let M be an interior point.
Prove that
min { MA, MB, MC } + MA + MB + MC < AB + BC + CA . 


212.

A set S of points in space has at least three elements
and satisfies the condition that, for any two distinct points A
and B in S, the right bisecting plane of the segment AB is a
plane of symmetry for S. Determine all possible finite sets
S that satisfy the condition.