PROBLEMS FOR OCTOBER 2004
Please send your solution to
Prof. Edward J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
no later than December 5, 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.

339.

Let a, b, c be integers with abc ¹ 0,
and u, v, w be integers, not all zero, for which
au^{2} + bv^{2} + cw^{2} = 0 . 

Let r be any rational number. Prove that the equation
ax^{2} + by^{2} + cz^{2} = r 

is solvable.

340.

The lock on a safe consists of three wheels, each
of which may be set in eight different positions. Because of a
defect in the safe mechanism, the door will open if any two of
the three wheels is in the correct position. What is the smallest
number of combinations which must be tried by someone not knowing
the correct combination to guarantee opening the safe?

341.

Let s, r, R respectively specify the semiperimeter,
inradius and circumradius of a triangle ABC.


(a) Determine a necessary and sufficient condition on s, r, R
that the sides a, b, c of the triangle are in arithmetic
progression.


(b) Determine a necessary and sufficient condition on s, r, R
that the sides a, b, c of the triangle are in geometric progression.

342.

Prove that there are infinitely many solutions in
positive integers of the system

343.

A sequence { a_{n} } of integers is defined by
a_{0} = 0 , a_{1} = 1 , a_{n} = 2a_{n1} + a_{n2} 

for n > 1. Prove that, for each nonnegative integer k,
2^{k} divides a_{n} if and only if 2^{k} divides n.

344.

A function f defined on the positive integers is
given by
f(1) = 1 , f(3) = 3 , f(2n) = f(n) , 

for each positive integer n. Determine, with proof, the number
of positive integers no exceeding 2004 for which f(n) = n.

345.

Let C be a cube with edges of length 2.
Construct a solid figure with fourteen faces by cutting off all
eight corners of C, keeping the new faces perpendicular to
the diagonals of the cuhe and keeping the newly formed faces identical.
If the faces so formed all have the same area, determine the common
area of the faces.