2002 BAMO

Part of BAMO Contests

Subcontests

(5)

1

2002 BAMO p5 prime-testing trail of Professor Moriarty, station 1429

Professor Moriarty has designed a “prime-testing trail.” The trail has

2002

stations, labeled

1,... , 2002

. Each station is colored either red or green, and contains a table which indicates, for each of the digits

0, ..., 9

, another station number. A student is given a positive integer

n

, and then walks along the trail, starting at station

1

. The student reads the first (leftmost) digit of

n,

and looks this digit up in station

1

’s table to get a new station location. The student then walks to this new station, reads the second digit of

n

and looks it up in this station’s table to get yet another station location, and so on, until the last (rightmost) digit of

n

has been read and looked up, sending the student to his or her final station. Here is an example that shows possible values for some of the tables. Suppose that

n = 19

: https://cdn.artofproblemsolving.com/attachments/f/3/db47f6761ca1f350e39d53407a1250c92c4b05.png Using these tables, station

1

1

leads to station

29

29

9

leads to station

1429

1429

is green. Professor Moriarty claims that for any positive integer

n

, the final station (in the example,

1429

) will be green if and only if

n

is prime. Is this possible?

1

2002 BAMO p2 tiling regular 10-gons,12-gons into rhombuses

In the illustration, a regular hexagon and a regular octagon have been tiled with rhombuses. In each case, the sides of the rhombuses are the same length as the sides of the regular polygon. (a) Tile a regular decagon (

10

-gon) into rhombuses in this manner. (b) Tile a regular dodecagon (

12

-gon) into rhombuses in this manner. (c) How many rhombuses are in a tiling by rhombuses of a

2002

-gon? Justify your answer. https://cdn.artofproblemsolving.com/attachments/8/a/8413e4e2712609eba07786e34ba2ce4aa72888.png

1

2002 BAMO p4 inequality with largest odd divisor

n \ge 1

a_n

be the largest odd divisor of

n

b_n = a_1+a_2+...+a_n

b_n \ge \frac{ n^2 + 2}{3}

, and determine for which

n

equality holds.
For example,

a_1 = 1, a_2 = 1, a_3 = 3, a_4 = 1, a_5 = 5, a_6 = 3

b_6 = 1 + 1 + 3 + 1 + 5 + 3 = 14 \ge \frac{ 6^2 + 2}{3}= 12\frac23

1

2002 BAMO p1 angle wanted, square share side with right triangle

ABC

be a right triangle with right angle at

B

ACDE

be a square drawn exterior to triangle

ABC

M

is the center of this square, find the measure of

\angle MBC

1

A hard game theory problem

A game is played with two players and an initial stack of

n

(n \geq 3)

. The players take turns choosing one of the stacks of pennies on the table and splitting it into two stacks. The winner is the player who makes a move that causes all stacks to be of height

1

2.

For which starting values of n does the player who goes first win, assuming best play by both players?