5

Part of 1998 Tournament Of Towns

Problems(8)

TOT 1998 Spring OJ5 divide isoceles into 3 triangles, new isosceles

Source:

Pinocchio claims that he can divide an isoceles triangle into three triangles, any two of which can be put together to form a new isosceles triangle. Is Pinocchio lying?
(A Shapovalov)

combinatorial geometrycombinatoricsgeometryisosceles

TOT 1998 Spring AJ5 square is divided into 25 small squares

Source:

A square is divided into

25

small squares. We draw diagonals of some of the small squares so that no two diagonals share a common point (not even a common endpoint). What is the largest possible number of diagonals that we can draw?
(I Rubanov)

Tilingcombinatorial geometrycombinatoricsSquaresdiagonals

circumcenter of ABC lies on the bisector of the original angle

Source: Tournament of Towns Spring 1998 Seniors O p5

A circle with center

O

is inscribed in an angle. Let

A

be the reflection of

O

across one side of the angle. Tangents to the circle from

A

intersect the other side of the angle at points

B

C

. Prove that the circumcenter of triangle

ABC

lies on the bisector of the original angle.
(I.Sharygin)

geometryangle bisectorreflectioncircle

TOT 1998 Spring AS5 good and bad labyrinths in an 8x8 chessboard

Source:

A "labyrinth" is an

8 \times 8

chessboard with walls between some neighboring squares. If a rook can traverse the entire board without jumping over the walls, the labyrinth is "good" ; otherwise it is "bad" . Are there more good labyrinths or bad labyrinths?
(A Shapovalov)

Chessboardcombinatorics

(n,m)-crocodile chess piece

Source: ToT Junior O Level Autumn 1998

n

m

be given positive integers. In one move, a chess piece called an

(n,m)

-crocodile goes

n

squares horizontally or vertically and then goes

m

squares in a perpendicular direction. Prove that the squares of an infinite chessboard can be painted in black and white so that this chess piece always moves from a black square to a white one or vice-versa.

analytic geometrycombinatorics proposedcombinatorics

TOT 1998 Autumn OS5 intelligence quotient (IQ) of a country problem

Source:

The intelligence quotient (IQ) of a country is defined as the average IQ of its entire population. It is assumed that the total population and individual IQs remain constant throughout. (a) (i) A group of people from country

A

has emigrated to country

B

. Show that it can happen that as a result , the IQs of both countries have increased. (ii) After this, a group of people from

B

, which may include immigrants from

A

A

. Can it happen that the IQs of both countries will increase again? (b) A group of people from country

A

has emigrated to country

B

, and a group of people from

B

has emigrated to country

C

. It is known that a s a result , the IQs o f all three countries have increased. After this, a group of people from

C

B

and a group of people from

B

A

. Can it happen that the IQs of all three countries will increase again?
(A Kanel, B Begun)

combinatoricsAverage

TOT 1998 Autumn AJ5 20 beads of 10 colours, 2 of each colour, in 10 boxes

Source:

20

10

colours, two of each colour. They are put in

10

boxes. It is known that one bead can be selected from each of the boxes so that each colour is represented. Prove that the number of such selections is a non-zero power of

2

Coloringcombinatorics

TOT 1998 Autumn AS5 rectangular parallelepiped contains another

Source:

The sum of the length, width, and height of a rectangular parallelepiped will be called its size. Can it happen that one rectangular parallelepiped contains another one of greater size?
(A Shen)

3D geometrygeometrygeometric inequalityparallelepiped