+/- 1 +/- 2 +/- 3 +/- 4 +/- ... +/- 96 = 1996.At most how many of the +/- signs can be replaced by a + sign?
Problem 2/19. We say that (a,b,c) is a primitive Heronian triple if a, b, and c are positive integers with no common factors (other than 1), and if the area of the triangle whose sides measure a, b, and c is also an integer. Prove that if a=96, then b and c must both be odd.
Problem 3/19. The numbers in the 7 by 8 rectangle shown below were obtained by putting together the 28 distinct dominoes of a standard set, recording the number of dots (ranging from 0 to 6) on each side of the dominoes, and then erasing the boundaries among them. Determine the original boundaries among the dominoes. (Note: each domino consists of two adjoint squares, referred to as its sides).
5 5 5 2 1 3 3 4 6 4 4 2 1 1 5 2 6 3 3 2 1 6 0 3 3 0 5 5 0 0 0 6 3 2 1 6 0 0 4 2 0 3 6 4 6 2 6 5 2 1 1 4 4 4 1 5Problem 4/19. Suppose that f satisfies the functional equation
2x + 29 2 f(x) + 3 f(-------) = 100 x + 80. x - 2
Find f(3).
Problem 5/19. In the figure below, determine the area of the shaded octagon as a fraction of the area of the square, where the boundaries of the octagon are lines drawn from the vertices of the square to the midpoints of the opposite sides.
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Solve as many of the problems as you can (you need not solve them all), and mail your solutions to:
Professor E. J. BarbeauMake sure that the front page of your solutions contains your full name and mailing address.
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3