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International Mathematical Talent Search

# Round 19

Problem 1/19. It is possible to replace each of the signs below by either - or + so that

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).

Problem 4/19. Suppose that f satisfies the functional equation

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.

Solve as many of the problems as you can (you need not solve them all), and mail your solutions to:

Professor E. J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
These problems are made available through the quarterly journal Mathematics and Informatics. Student subscriptions at US\$12 (student rate) or US\$18 (teacher rate) may be ordered from Prof. George Berzsenyi, Department of Mathematics, Box 121, Rose-Hulman Institute of Technology, Terre Haute, IN 47803-3999, U.S.A. Remittances should be payable to Mathematics and Informatics.