CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
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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 ,, xn be integers for which

x1 2 + x2 2 ++ xn 2 + n3 (2n-1)( x1 + x2 ++ xn )+ n2 .

Show that
(a) x1 , x2 ,, xn are all nonnegative;
(b) x1 + x2 ++ 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 DAE= 15ˆ . Let F be a point in AB with EFAB. It is known that EF= 1 2 AB and AD=a. Find the measure of the angle EAC and the length of the segment EC.


58.
Find integers a, b, c such that a0 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-BDAB-CD.



60.
Let n2 be an integer and M={1,2,,n}. For every integer k with 1kn-1, let

xk ={minA+maxA:AM,Ahaskelements}

where min A is the smallest and max A is the largest number in A. Determine k=1 n(-1)k-1 xk .


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