2015 BAMO

Part of BAMO Contests

Subcontests

(5)

1

Identical cube arrangements

n

identical cubes, each of size

1\times 1\times 1

. We arrange all of these

n

cubes to produce one or more congruent rectangular solids, and let

B(n)

be the number of ways to do this.
For example, if

n=12

, then one arrangement is twelve

1\times1\times1

cubes, another is one

3\times 2\times2

solid, another is three

2\times 2\times1

solids, another is three

4\times1\times1

solids, etc. We do not consider, say,

2\times2\times1

1\times2\times2

to be different; these solids are congruent. You may wish to verify, for example, that

B(12) =11

.
Find, with proof, the integer

m

10^m<B(2015^{100})<10^{m+1}

2

Perpendicular diagonals

In a quadrilateral, the two segments connecting the midpoints of its opposite sides are equal in length. Prove that the diagonals of the quadrilateral are perpendicular.
(In other words, let

M,N,P,

Q

be the midpoints of sides

AB,BC,CD,

DA

in quadrilateral

ABCD

. It is known that segments

MP

NQ

are equal in length. Prove that

AC

BD

are perpendicular.)

Cube problem

A

be a corner of a cube. Let

B

C

the midpoints of two edges in the positions shown on the figure below: http://i.imgur.com/tEODnV0.png The intersection of the cube and the plane containing

A,B,

C

is some polygon,

P

.
[*] How many sides does

P

have? Justify your answer. [*] Find the ratio of the area of

P

\triangle{ABC}

and prove that your answer is correct.

2

Powers of three

k

be a positive integer. Prove that there exist integers

x

y

, neither of which is divisible by

3

x^2+2y^2 = 3^k

Which is larger

Which number is larger,

A

B

A = \dfrac{1}{2015} (1 + \dfrac12 + \dfrac13 + \cdots + \dfrac{1}{2015})

B = \dfrac{1}{2016} (1 + \dfrac12 + \dfrac13 + \cdots + \dfrac{1}{2016}) \text{ ?}

Prove your answer is correct.

1

Various committees

Members of a parliament participate in various committees. Each committee consists of at least

2

people, and it is known that every two committees have at least one member in common. Prove that it is possible to give each member a colored hat (hats are available in black, white or red) so that every committee contains at least

2

members with different hat colors.

1

Boxes and cards

7

boxes arranged in a row and numbered

1

7

. You have a stack of

2015

cards, which you place one by one in the boxes. The first card is placed in box #

1

, the second in box #

2

, and so forth up to the seventh card which is placed in box #

7

. You then start working back in the other direction, placing the eighth card in box #

6

, the ninth in box #

5

, up to the thirteenth card being placed in box #

1

. The fourteenth card is then placed in box #

2

, and this continues until every card is distributed. What box will the last card be placed in?