MathDB

1

interchanging 3 piles by a sequence of moves

Three numbered tiles are arranged in a tray as shown: https://cdn.artofproblemsolving.com/attachments/d/0/d449364f92b7fae971fd348a82bafd25aa8ea1.jpg Show that we cannot interchange the

1

and the

3

by a sequence of moves where we slide a tile to the adjacent vacant space.

6

1

5 spheres of radius r are inside a right circular cone

Five spheres of radius

r

are inside a right circular cone. Four of the spheres lie on the base of the cone. Each touches two of the others and the sloping sides of the cone. The fifth sphere touches each of the other four and also the sloping sides of the cone. Find the volume of the cone.

5

1

construct segment (a^4 + b^4)^{1/4}

Show how to construct a line segment length

(a^4 + b^4)^{1/4}

given segments lengths

a

and

b

.

3

1

squares have their vertices in a (k+1)x(k+1) array of lattice points

S

is a

(k+1) \times (k+1)

array of lattice points. How many squares have their vertices in

S

?

2

1

integers in the form n = 2^b(2c+1), odd parts and sequence

Any positive integer

n

can be written in the form

n = 2^b(2c+1)

. We call

2c+1

the odd part of

n

. Given an odd integer

n > 0

, define the sequence

a_0, a_1, a_2, ...

as follows:

a_0 = 2^n-1, a_{k+1}

is the odd part of

3a_k+1

. Find

a_n

.

1

triangle of incenter and 2 excenters is similar to original triangle

The angles of the triangle

ABC

satisfy

\angle A / \angle C = \angle B / \angle A = 2

. The incenter is

O. K, L

are the excenters of the excircles opposite

B

and

A

respectively. Show that triangles

ABC

and

OKL

are similar.

1982 Brazil National Olympiad

Subcontests

interchanging 3 piles by a sequence of moves

5 spheres of radius r are inside a right circular cone

construct segment (a^4 + b^4)^{1/4}

squares have their vertices in a (k+1)x(k+1) array of lattice points

integers in the form n = 2^b(2c+1), odd parts and sequence

triangle of incenter and 2 excenters is similar to original triangle