2011 BAMO

Part of BAMO Contests

Subcontests

(5)

1

2011 BAMO 7 Pascal Triange a,b,c,d with b=2a and d=2c

Does there exist a row of Pascal’s Triangle containing four distinct values

a,b,c

d

b = 2a

d = 2c

? Recall that Pascal’s triangle is the pattern of numbers that begins as follows https://cdn.artofproblemsolving.com/attachments/2/1/050e56f0f1f1b2a9c78481f03acd65de50c45b.png where the elements of each row are the sums of pairs of adjacent elements of the prior row. For example,

10 =4+6

. Also note that the last row displayed above contains the four elements

a = 5,b = 10,d = 10,c = 5

b = 2a

d = 2c

, but these four values are NOT distinct.

2

2011 BAMO 3 Rubik cube 8x8x8, each face painted with different color

Consider the 8\times 8\times 8 Rubik’s cube below. Each face is painted with a different color, and it is possible to turn any layer, as you can with smaller Rubik’s cubes. Let

X

denote the move that turns the shaded layer shown (indicated by arrows going from the top to the right of the cube) clockwise by

90

degrees, about the axis labeled

X

X

is performed, the only layer that moves is the shaded layer. Likewise, define move

Y

to be a clockwise

90

-degree turn about the axis labeled Y, of just the shaded layer shown (indicated by the arrows going from the front to the top, where the front is the side pierced by the

X

rotation axis). Let

M

denote the move “perform

X

Y

.” https://cdn.artofproblemsolving.com/attachments/e/f/951ea75a3dbbf0ca23c45cd8da372595c2de48.png Imagine that the cube starts out in “solved” form (so each face has just one color), and we start doing move

M

repeatedly. What is the least number of repeats of

M

in order for the cube to be restored to its original colors?

2011 BAMO 5 elements of set are arranged in arithmetic progression

S

be a finite, nonempty set of real numbers such that the distance between any two distinct points in

S

is an element of

S

. In other words,

|x-y|

S

x \ne y

x

y

S

. Prove that the elements of

S

may be arranged in an arithmetic progression. This means that there are numbers

a

d

S = \{a, a+d, a+2d, a+3d, ..., a+kd, ...\}

2

2011 BAMO 4 symmetric point construction using only straightedge

In a plane, we are given line

\ell

A

B

neither of which lies on line

\ell

, and the reflection

A_1

A

\ell

. Using only a straightedge, construct the reflection

B_1

B

\ell

. Prove that your construction works.
Note: “Using only a straightedge” means that you can perform only the following operations: (a) Given two points, you can construct the line through them. (b) Given two intersecting lines, you can construct their intersection point. (c) You can select (mark) points in the plane that lie on or off objects already drawn in the plane. (The only facts you can use about these points are which lines they are on or not on.)

2011 BAMO 6 |AO|/|AA'|+ |BO|/|BB'|+|CO|/|CC'|= 1, 3 circle related

k_1, k_2

k_3

intersect in point

O

A, B

C

be the second intersection points (other than

O

k_2

k_3, k_1

k_3

k_1

k_2

, respectively. Assume that

O

lies inside of the triangle

ABC

AO,BO

CO

intersect circles

k_1, k_2

k_3

for a second time at points

A', B'

C'

, respectively. If

|XY|

denotes the length of segment

XY

\frac{|AO|}{|AA'|}+\frac{|BO|}{|BB'|}+\frac{|CO|}{|CC'|}= 1

1

2011 BAMO 2 numbers in 5 circles in a row

Five circles in a row are each labeled with a positive integer. As shown in the diagram, each circle is connected to its adjacent neighbor(s). The integers must be chosen such that the sum of the digits of the neighbor(s) of a given circle is equal to the number labeling that point. In the example, the second number

23 = (1+8)+(5+9)

, but the other four numbers do not have the needed value. https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMi9lL2M2MzVkMmMyYTRlZjliNWEzYWNkOTM2OGVmY2NkOGZmOWVkN2VmLnBuZw==&rn=MjAxMSBCQU1PIDIucG5n What is the smallest possible sum of the five numbers? How many possible arrangements of the five numbers have this sum? Justify your answers.

1

2011 BAMO 1 set of identical square tiles with side length 1

A set of identical square tiles with side length

1

is placed on a (very large) floor. Every tile after the first shares an entire edge with at least one tile that has already been placed. - What is the largest possible perimeter for a figure made of

10

tiles? - What is the smallest possible perimeter for a figure made of

10

tiles? - What is the largest possible perimeter for a figure made of

2011

tiles? - What is the smallest possible perimeter for a figure made of

2011

tiles? Prove that your answers are correct.