Navigation Panel: Previous | Up | Forward | Graphical Version | PostScript version | U of T Math Network Home

# Round 19

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

```        *********************************
***            **             ***
* *  *        *  *         *  * *
*  *     *   *    *     *    *  *
*   *       *      **        *  *
*   *       *   *   *       *   *
*    *     * *.... **      *    *
*     *   * ........ **   *     *
*     ** *........... *   *     *
*  *   * *............. **   *  *
*       *.............. *       x
*  *   * *.............* *   *  *
*     *   *............*  *     *
*     *   *.......... *    *    *
*    *     * * .. *  *     *    *
*   *       *  **   *       *   *
*  *        **      *        *  *
*  *    *    *     *   *      * *
* *  *        *   *        *  * *
***            * *            ***
*********************************
```

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