PROBLEMS FOR APRIL
Please send your solution to
Edward J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
no later than May 31, 2003.
It is important that your complete mailing address
and your email address appear on the front page.
Notes. Let x be a real number. The
inverse
tangent function, tan
^{1} x
(sometimes referred to as arctanx) is that number
q
for which
p/2 <
q <
p/2 and tan
q = x.

220.

Prove or disprove: A quadrilateral with one pair
of opposite sides and one pair of opposite angles equal is
a parallelogram.

221.

A cycloid is the locus of a point P fixed on
a circle that rolls without slipping upon a line u. It consists
of a sequence of arches, each arch extending from that position
on the locus at which the point P rests on the line u,
through a curve that rises to a position whose distance from
u is equal to the diameter of the generating circle and then
falls to a subsequent position at which P rests on the line u.
Let v be the straight line parallel to u that is tangent to
the cycloid at the point furthest from the line u.


(a) Consider a position of the generating circle, and let P
be on this circle and on the cycloid. Let PQ be the chord on this
circle that is parallel to u (and to v). Show that the locus
of Q is a similar cycloid formed by a circle of the same radius
rolling (upside down) along the line v.


(b) The region between the two cycloids consists of a number
of ``beads''. Argue that the area of one of these beads is equal to
the area of the generating circle.


(c) Use the considerations of (a) and (b) to find the
area between u and one arch of the cycloid using a method
that does not make use of calculus.

222.

Evaluate

¥ å
n=1

tan^{1} 
æ ç
è


2 n^{2}


ö ÷
ø

. 


223.

Let a, b, c be positive real numbers for which
a + b + c = abc. Prove that

224.

For x > 0, y > 0, let g(x, y) denote the
minimum of the three quantities, x, y + 1/x and 1/y.
Determine the maximum value of g(x, y) and where this
maximum is assumed.

225.

A set of n lighbulbs, each with an onoff
switch, numbered 1, 2, ¼, n are arranged in a line.
All are initially off. Switch 1 can be operated at any time
to turn its bulb on of off. Switch 2 can turn bulb 2 on or
off if and only if bulb 1 is off; otherwise, it does not function.
For k ³ 3, switch k can turn bulb k on or off if and only
if bulb k1 is off and bulbs 1, 2, ¼, k2 are all on;
otherwise it does not function.


(a) Prove that there is an algorithm that will turn all of the
bulbs on.


(b) If x_{n} is the length of the shortest algorithm that will
turn on all n bulbs when they are initially off, determine the
largest prime divisor of 3x_{n} + 1 when n is odd.

226.

Suppose that the polynomial f(x) of degree n ³ 1
has all real roots and that l > 0. Prove that the set
{ x Î R : f(x)  £ lf¢(x) }
is a finite union of closed intervals whose total length
is equal to 2nl.