International Mathematical Talent Search

+/- 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 5

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

Department of Mathematics

University of Toronto

Toronto, ON M5S 3G3

These problems are made available through the quarterly journal

This page last updated: February 3, 1997

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