PROBLEMS FOR AUGUST
Please send your solution to
Edward J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
no later than September 30, 2003.
It is important that your complete mailing address
and your email address appear on the front page.

227.

[Since the original statement of this problem
in May was incorrect and not everyone picked up the correction,
it is reposed.] Let n be an integer exceeding 2 and let
a_{0}, a_{1}, a_{2}, ¼, a_{n}, a_{n+1} be positive real numbers for which
a_{0} = a_{n}, a_{1} = a_{n+1} and
a_{i1} + a_{i+1} = k_{i} a_{i} 

for some positive integers k_{i}, where 1 £ i £ n.


Prove that
2n £ k_{1} + k_{2} + ¼+ k_{n} £ 3n . 


241.

[Corrected.] Determine
sec40^{°} + sec80^{°} + sec160^{°} . 


248.

Find all real solutions to the equation

249.

The nonisosceles right triangle ABC has
ÐCAB = 90^{°}. Its inscribed circle with centre
T touches the sides AB and AC at U and V respectively.
The tangent through A of the circumscribed circle of
triangle ABC meets UV
in S. Prove that:
(a) ST  BC;


(b) d_{1}  d_{2}  = r , where
r is the radius of the inscribed circle, and d_{1} and d_{2} are the
respective distances from S to AC and AB.

250.

In a convex polygon \frakP, some diagonals
have been drawn so that no two have an intersection in the interior
of \frakP. Show that there exists at least two vertices of
\frak P, neither of which is an enpoint of any of these diagonals.

251.

Prove that there are infinitely many positive integers
n for which the numbers { 1, 2, 3, ¼, 3n } can be
arranged in a rectangular array with three rows and n columns for which
(a) each row has the same sum, a multiple of 6, and (b) each
column has the same sum, a multiple of 6.

252.

Suppose that a and b are the roots of the
quadratic x^{2} + px + 1 and that c and d are the roots of
the quadratic x^{2} + qx + 1. Determine
(a  c)(b  c)(a + d)(b + d) as a function of p and q.

253.

Let n be a positive integer and let q = p/(2n+1). Prove that cot^{2} q, cot^{2} 2q,
¼, cot^{2} nq are the solutions of the equation

æ è

2n + 1
1

ö ø

x^{n}  
æ è

2n + 1
3

ö ø

x^{n1} + 
æ è

2n+1
5

ö ø

x^{n2} ¼ = 0 . 


254.

Determine the set of all triples (x, y, z) of integers
with 1 £ x, y, z £ 1000 for which x^{2} + y^{2} + z^{2} is a
multiple of xyz.