PROBLEMS FOR JULY AND AUGUST |
Because of the variability of summer plans, the usual ration
of problems has been doubled and the deadline set later so that
students can have a chance to organize their work conveniently.
Send your solutions to Prof. E.J. Barbeau, Department of Mathematics,
University of Toronto, Toronto, ON M5S 3G3 no later than
September 10, 2002. Please make sure that the front page of
your solution contains your complete mailing address and your email
address.
Notes. A composite integer is one that has positive
divisors other than 1 and itself; it is not prime. A set of
point in the plane is concyclic (or cyclic, inscribable)
if and only if there is a circle that passes through all of them.
-
157.
-
Prove that if the quadratic equation
has nonzero integer solutions, then
is a composite
integer.
-
158.
-
Let
be a polynomial with real coefficients for
which the equation
has no real solution. Prove that the
equation
has no real solution either.
-
159.
-
Let
. Prove that the area of the
bounded region enclosed by the curves with equations
and
cannot exceed
.
-
160.
-
Let
be the incentre of the triangle
and
be the point of contact of the inscribed circle with the side
. Suppose that
is produced outside of the triangle
to
so that the length
is equal to the semi-perimeter
of
. Prove that the quadrilateral
is concyclic if and only if angle
is equal to
.
-
161.
-
Let
be positive real numbers for which
. Prove that
-
162.
-
Let
and
be fixed points in the plane.
Find all positive integers
for which the following assertion
holds:
-
-
among all triangles
with
, the one with the largest area is isosceles.
-
163.
-
Let
and
re the respective circumradius and
inradius of triangle
(
). Prove that, if
and
, then the two
triangles are similar.
-
164.
-
Let
be a positive integer and
a set with
distinct elements. Suppose that there are
distinct subsets
of
for which the union of any four contains no more that
elements. Prove that
.
-
165.
-
Let
be a positive integer. Determine all
tples
of positive integers for which
and there is no subset of them
whose sum is equal to
.
-
166.
-
Suppose that
is a real-valued function defined on
the reals for which
for all real
and
. Prove that
for all real
.
-
167.
-
Let
and
. Prove
that, for each positive integer
,
.
-
168.
-
Determine the value of
-
169.
-
Prove that, for each positive integer
exceeding 1,
-
170.
-
Solve, for real
,