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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 xk.

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