PROBLEMS FOR JANUARY
Solutions should be submitted to
Prof. E.J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
no later than February 28, 2001
- 55.
-
A textbook problem has the following
form: A man is standing in a line in front of
a movie theatre. The fraction x of the line is
in front of him, and the fraction y of the
line is behind him, where x and y are rational
numbers written in lowest terms. How many people
are there in the line? Prove that, if the problem
has an answer, then that answer must be the least
common multiple of the denominators of x and
y.
- 56.
-
Let n be a positive integer and let
x1, x2, ¼, xn be integers for which
|
x12 + x22 + ¼+ xn2 + n3 £ (2n-1)(x1 + x2 + ¼+ xn) + n2 . |
|
Show that
-
-
(a) x1, x2, ¼, xn are all nonnegative;
-
-
(b) x1 + x2 + ¼+ xn + n + 1 is not
a perfect square.
- 57.
-
Let ABCD be a rectangle and let E be a point
in the diagonal BD with ÐDAE = 15°. Let
F be a point in AB with EF ^AB. It is known that
EF = 1/2AB and AD = a. Find the measure of the
angle ÐEAC and the length of the segment EC.
- 58.
-
Find integers a, b, c such that
a ¹ 0 and the
quadratic function f(x) = ax2 + bx + c satisfies
|
f(f(1)) = f(f(2)) = f(f(3)) . |
|
- 59.
-
Let ABCD be a concyclic quadrilateral.
Prove that
|
|AC - BD | £ |AB - CD | . |
|
- 60.
-
Let n ³ 2 be an integer and
M = { 1, 2, ¼, n }. For every integer k
with 1 £ k £ n-1, let
|
xk = |
å
| { min A + max A :A Í M, A has k elements } |
|
where min A is the smallest and max A is the largest
number in A.
Determine
åk = 1n (-1)
k-1 x
k.
