PROBLEMS FOR JULY
Please send your solution to
Edward J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
no later than September 10, 2003.
It is important that your complete mailing address
and your email address appear on the front page.
Notes. A partition of the positive integer n is
a representation (up to order) of n as a sum of not necessarily
distinct positive integers, i.e.,
n = a_{1} + a_{2} + ¼+ a_{k} with a_{1} ³ a_{2} ³ ¼ ³ a_{k} ³ 1. The number of distinct partitions is
denoted by p(n). Thus, p(6) = 11 since 6 can be written as
6 = 5 + 1 = 4 + 2 = 4 + 1 + 1 = 3 + 3 = 3 + 2 + 1 = 3 + 1 + 1 + 1 = 2 + 2 + 2 = 2 + 2 + 1 + 1 = 2 + 1 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 + 1.

241.

Determine sec40^{°} + sec80^{°}+ sec169^{°}.

242.

Let ABC be a triangle with sides of length a,
b, c oppposite respective angles A, B, C. What is the
radius of the circle that passes through the points A, B and
the incentre of triangle ABC when angle C is equal to
(a) 90^{°}; (b) 120^{°}; (c) 60^{°}. (With
thanks to Jean Turgeon, Université de Montréal.)

243.

The inscribed circle, with centre I, of the
triangle ABC touches the sides BC, CA and AB at the
respective points D, E and F. The line through A
parallel to BC meets DE and DF produced at the respective
points M and N. The modpoints of DM and DN are
P and Q respectively. Prove that A, E, F, I, P, Q lie on
a common circle.

244.

Let x_{0} = 4, x_{1} = x_{2} = 0, x_{3} = 3, and,
for n ³ 4, x_{n+4} = x_{n+1} + x_{n}. Prove that, for
each prime p, x_{p} is a multiple of p.

245.

Determine all pairs (m, n) of positive integers
with m £ n for which an m ×n rectangle can be tiles
with congrent pieces formed by removing a 1 ×1 square
from a 2 ×2 square.

246.

Let p(n) be the number of partitions of the positive
integer n, and let q(n) denote the number of finite sets
{ u_{1}, u_{2}, u_{3}, ¼, u_{k} } of positive integers that
satisfy u_{1} > u_{2} > u_{3} > ¼ > u_{k} such that
n = u_{1} + u_{3} + u_{5} + ¼ (the sum of the ones with odd
indices). Prove that p(n) = q(n) for each positive integer n.


For example, q(6) counts the sets { 6 },
{ 6, 5 }, { 6, 4 }, { 6, 3 }, { 6. 2 }, { 6, 1 },
{ 5, 4, 1 }, { 5, 3, 1 }, { 5, 2, 1 }, { 4, 3, 2 },
{ 4, 3, 2, 1 }.

247.

Let ABCD be a convex quadrilateral with no pairs of
parallel sides. Associate to side AB a point T as follows.
Draw lines through A and B parallel to the opposite side CD.
Let these lines meet CB produced at B¢ and DA produced at
A¢, and let T be the intersection of AB and B¢A¢. Let
U, V, W be points similarly constructed with respect to sides
BC, CD, DA, respectively. Prove that TUVW is a parallelogram.