International Mathematical Talent Search

**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. 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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