2

Part of 2009 Tournament Of Towns

Problems(7)

Forty weights - TT 2009 Junior-O2

Source:

There are forty weights:

1, 2, \cdots , 40

grams. Ten weights with even masses were put on the left pan of a balance. Ten weights with odd masses were put on the right pan of the balance. The left and the right pans are balanced. Prove that one pan contains two weights whose masses differ by exactly

20

grams.
(4 points)

modular arithmetic

Unit Cubes - TT 2009 Junior-A2

Source:

1000

unit cubes. Each has

2

opposite red faces,

2

opposite blue faces and

2

opposite white faces. Mike assembles them into a

10 \times 10 \times 10

cube. Whenever two unit cubes meet face to face, these two faces have the same colour. Prove that an entire face of the

10 \times 10 \times 10

cube has the same colour.
(6 points)

geometry3D geometrycombinatorics unsolvedcombinatorics

Cutting a rectangle - TT 2009 Senior-A2

Source:

A non-square rectangle is cut into

N

rectangles of various shapes and sizes. Prove that one can always cut each of these rectangles into two rectangles so that one can construct a square and rectangle, each figure consisting of

N

pieces.
(6 points)

geometryrectangleratiogeometry unsolved

All six points are in the same plane - TT 2009 Senior-O2

Source:

A; B; C; D; E

F

are points in space such that

AB

DE

BC

EF

CD

FA

AB \neq DE

. Prove that all six points lie in the same plane.
(4 points)

geometryparallelogram

2009 ToT Spring Senior O P2 even number of segments

Source:

Several points on the plane are given, no three of them lie on the same line. Some of these points are connected by line segments. Assume that any line that does not pass through any of these points intersects an even number of these segments. Prove that from each point exits an even number of the segments.

combinatorial geometrycombinatorics

2009 ToT Spring Junior A P2 cut a polygon, ratio 1:2 related

Source:

(a) Find a polygon which can be cut by a straight line into two congruent parts so that one side of the polygon is divided in half while another side at a ratio of

1 : 2

. (b) Does there exist a convex polygon with this property?

2009 ToT Spring Junior O P2 7^7^7^7^7^7^7, digits

Source:

a^b

denote the number

ab

. The order of operations in the expression 7^7^7^7^7^7^7 must be determined by parentheses (

5

pairs of parentheses are needed). Is it possible to put parentheses in two distinct ways so that the value of the expression be the same?

number theoryDigits