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# Round 16

Problem 1/16. Prove that if a + b + c = 0, then a^3 + b^3 + c^3 = 3abc.

Problem 2/16. For a positive integer n, let P(n) be the product of the nonzero base 10 digits of n. Call n "prodigitious" if P(n) divides n. Show that one cannot have a sequence of fourteen consecutive positive integers that are all prodigitious.

Problem 3/16. Disks numbered 1 through n are placed in a row of squares, with one square left empty. A move consists of picking up one of the disks and moving it into the empty square, with the aim to rearrange the disks in the smallest number of moves so that disk 1 is in square 1, disk 2 is in square 2, and so on until disk n is in square n and the last square is left empty. For example, if the original arrangement is

```        +---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   |   |   |   |   |   |
| 3 | 2 | 1 | 6 | 5 | 4 |   | 9 | 8 | 7 | 12| 11| 10|
|   |   |   |   |   |   |   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+
```

then it takes at least 14 moves; e.g., we could move the disks into the empty square in the following order: 7, 10, 3, 1, 3, 6, 4, 6, 9, 8, 9, 12, 11, 12.

What initial arrangement requires the largest number of moves if n = 1995? Specify the number of moves required.

Problem 4/16. Let ABCD be an arbitrary convex quadrilateral, with E, F, G, H the midpoints of its sides, as shown in the figure below. Prove that one can piece together triangles AEH, BEF, CFG, DGH to form a parallelogram congruent to parallelogram EFGH.

```                             C
*
G    *    *
o         *
* o   o       *
D *   o        o    *
o             o  *
* o                  oF
o                  o  *
Ho                  o    *
o              o       *
*      o         o         *
o   o            *
***************o***************
A               E               B
```

Problem 5/16. An equiangular polygon ABCDEFGH has sides of length 2, 2 sqrt(2), 4, 4 sqrt(2), 6, 7, 7, 8. Given that AB = 8, find the length of EF.

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
Make sure that the front page of your solutions contains your full name and mailing address.
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.
This page last updated: February 3, 1997
Original Web Site Creator / Mathematical Content Developer: Philip Spencer
Current Network Coordinator and Contact Person: Joel Chan - mathnet@math.toronto.edu

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