PROBLEMS FOR DECEMBER
Please send your solution to
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
no later than January 15, 2004.
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and your email address appear on the front page.
Let a, b, c be the lengths of the sides
of a triangle and let
s = 1/2(a + b + c) be its semi-perimeter and r
be the radius of the inscribed circle. Prove that
and indicate when equality holds.
(s - a)-2 + (s - b)-2 + (s - c)-2 ³ r-2|
Let m and n be positive integers for which
m < n. Suppose that an arbitrary set of n integers is
given and the following operation is performed: select any
m of them and add 1 to each. For which pairs (m, n) is
it always possible to modify the given set by performing the
operation finitely often to obtain a set for which all the
integers are equal?
(a) Show that 4mn - m - n can be an integer
square for infinitely many pairs (m, n) of integers.
Is it possible for either m or n to be positive?
(b) Show that there are infinitely many pairs
(m, n) of positive integers for which 4mn - m - n is
one less than a perfect square.
(a) For which values of n is it possible to
construct a sequence of abutting segments in the plane to form
a polygon whose side lengths are 1, 2, ¼, n exactly in
this order, where two neighbouring segments are perpendicular?
(b) For which values of n is it possible to construct
a sequence of abutting segments in space to form a polygon whose
side lengths are 1, 2, ¼, n exactly in this order, where
any two of three sucessive segments are perpendicular?
Consider all finite sequences of positive integers
whose sum is n. Determine T(n, k), the number of times that
the positive integer k occurs in all of these sequences taken
Let a be the result of tossing a black die
(a number cube whose sides are numbers from 1 to 6 inclusive),
and b the result of tossing a white die. What is the
probability that there exist real numbers x, y, z for which
x + y + z = a and xy + yz + zx = b?
Suppose that at the vertices of a pentagon
five integers are specified in such a way that the sum of
the integers is positive. If not all the integers are
non-negative, we can perform the following operation:
suppose that x, y, z are three consecutive integers
for which y < 0; we replace them respectively by
the integers x + y, -y, z + y. In the event that
there is more than one negative integer, there is a
choice of how this operation may be performed.
Given any choice of integers, and any sequence of
operations, must we arrive at a set of nonnegative
integers after a finite number of steps?
For example, if we start with the numbers (2, -3, 3, -6, 7)
around the pentagon,
we can produce (1, 3, 0, -6, 7) or (2, -3, -3, 6, 1).