PROBLEMS FOR NOVEMBER, 2004
Please send your solutions to
Prof. Edward J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
no later than December 31, 2004.
It is important that your complete mailing address
and your email address appear on the front page.
If you do not write your family name last, please
underline it.
Notes. A realvalued function f(x) of a real variable is
increasing if and only if u < v implies that f(u) £ f(v).
The circumcircle of a triangle is that circle that passes
through its three vertices; its centre is the circumcentre
of the triangle. The incircle of a triangle is that circle
that is tangent internally to its three sides; its centre is the
incentre of the triangle.

346.

Let n be a positive integer. Determine the
set of all integers that can be written in the form
where a_{1}, a_{2}, ¼, a_{n} are all positive integers.

347.

Let n be a positive integer and { a_{1}, a_{2}, ¼, a_{n} } a finite sequence of real numbers which contains
at least one positive term. Let S be the set of indices k
for which at least one of the numbers
a_{k}, a_{k} + a_{k+1}, a_{k} + a_{k+1} + a_{k+2}, ¼,a_{k} + a_{k+1} + ¼+ a_{n} 

is positive. Prove that

å
 { a_{k} : k Î S } > 0 . 


348.

Suppose that f(x) is a realvalued function defined
for real values of x. Suppose that f(x)  x^{3} is an increasing
function. Must f(x)  x  x^{2} also be increasing?

349.

Let s be the semiperimeter of triangle ABC.
Suppose that L and N are points on AB and CB produced
(i.e., B lies on segments AL and CL) with
AL  = CN  = s. Let K be the point
symmetric to B with respect to the centre of the circumcircle
of triangle ABC. Prove that the perpendicular from K to the
line NL passes through the incentre of triangle ABC.

350.

Let ABCDE be a pentagon inscribed in a circle with
centre O. Suppose that its angles are given by
ÐB = ÐC = 120^{°}, ÐD = 130^{°},
ÐE = 100^{°}. Prove that BD, CE and AO are
concurrent.

351.

Let { a_{n} } be a sequence of real numbers for which
a_{1} = 1/2 and, for n ³ 1,
a_{n+1} = 
a_{n}^{2}
a_{n}^{2}  a_{n} + 1

. 

Prove that, for all n, a_{1} + a_{2} + ¼+ a_{n} < 1.

352.

Let ABCD be a unit square with points M and N
in its interior. Suppose, further, that MN produced does not
pass through any vertex of the square. Find the smallest value of
k for which, given any position of M and N, at least one
of the twenty triangles with vertices chosen from the set
{ A, B, C, D, M, N } has area not exceeding k.