2006 BAMO

Part of BAMO Contests

Subcontests

(5)

1

2006 BAMO p1 rearranging n^2 chairs in a nxn square classroom

All the chairs in a classroom are arranged in a square

n\times n

array (in other words,

n

n

rows), and every chair is occupied by a student. The teacher decides to rearrange the students according to the following two rules: (a) Every student must move to a new chair. (b) A student can only move to an adjacent chair in the same row or to an adjacent chair in the same column. In other words, each student can move only one chair horizontally or vertically. (Note that the rules above allow two students in adjacent chairs to exchange places.) Show that this procedure can be done if

n

is even, and cannot be done if

n

1

2006 BAMO p5 k switces in a row, 4 directions each

k

switches arranged in a row, and each switch points up, down, left, or right. Whenever three successive switches all point in different directions, all three may be simultaneously turned so as to point in the fourth direction. Prove that this operation cannot be repeated infinitely many times.

1

2006 BAMO p4 squares of infinite square grid colored grey and white

n

squares of an infinite square grid are colored grey, and the rest are colored white. At each step, a new grid of squares is obtained based on the previous one, as follows. For each location in the grid, examine that square, the square immediately above, and the square immediately to the right. If there are two or three grey squares among these three, then in the next grid, color that location grey, otherwise, color it white. Prove that after at most n steps all the squares in the grid will be white. Below is an example with

n = 4

. The first grid shows the initial configuration, and the second grid shows the configuration after one step. https://cdn.artofproblemsolving.com/attachments/1/a/87f7e3892cdb45fb3529127234aae2cea08749.png

1

2006 BAMO p3 P is centroid of ABC iff P is centroid of A_1B_1C_1

ABC

A_1

BC

B_1

CA

C_1

AB

in such a way that the three segments

AA_1, BB_1

CC_1

intersect in one point

P

P

is the centroid of triangle

ABC

P

is the centroid of triangle

A_1B_1C_1

.
Note: A median in a triangle is a segment connecting a vertex of the triangle with the midpoint of the opposite side. The centroid of a triangle is the intersection point of the three medians of the triangle. The centroid of a triangle is also known by the names ”center of mass” and ”medicenter” of the triangle.

1

2006 BAMO p2 2005, 2006 as sum of at least 2 consecutive odd pos. int.

24 = 3+5+7+9

24

can be written as the sum of at least two consecutive odd positive integers. (a) Can

2005

be written as the sum of at least two consecutive odd positive integers? If yes, give an example of how it can be done. If no, provide a proof why not. (b) Can

2006

be written as the sum of at least two consecutive odd positive integers? If yes, give an example of how it can be done. If no, provide a proof why not.