MathDB

2

Each positive integer shall be coloured red or green

Each positive integer shall be coloured red or green such that it satisfies the following properties: - The sum of three not necessarily distinct red numbers is a red number. - The sum of three not necessarily distinct green numbers is a green number. - There are red and green numbers. Find all such colorations!

Hojoo and Kestutis make moves in turn

At the start of the game there are

r

red and

g

green pieces/stones on the table. Hojoo and Kestutis make moves in turn. Hojoo starts. The person due to make a move, chooses a colour and removes

k

pieces of this colour. The number

k

has to be a divisor of the current number of stones of the other colour. The person removing the last piece wins. Who can force the victory?

3

2

German angle bisecting problem

In triangle

ABC

points

E

and

F

lie on sides

AC

and

BC

such that segments

AE

and

BF

have equal length, and circles formed by

A,C,F

and by

B,C,E,

respectively, intersect at point

C

and another point

D.

Prove that that the line

CD

bisects

\angle ACB.

Remove four points from R^3

A set

E

of points in the 3D space let

L(E)

denote the set of all those points which lie on lines composed of two distinct points of

E.

Let

T

denote the set of all vertices of a regular tetrahedron. Which points are in the set

L(L(T))?

1

2

A regular polygon with 2007 vertices

Consider a regular polygon with 2007 vertices. The natural numbers

1,2, \ldots, 4014

shall be distributed across the vertices and edge midpoints such that for each side the sum of its three numbers (two end points, one side center) has the same value. Show that such an assignment is possible.

Linear and quadratic sum of digits property

For which numbers

n

is there a positive integer

k

with the following property: The sum of digits for

k

is

n

and the number

k^2

has sum of digits

n^2.

4

2

Solve this equation.

Let

a

be a positive integer. How many non-negative integer solutions x does the equation

\lfloor \frac{x}{a}\rfloor = \lfloor \frac{x}{a+1}\rfloor

have?

\lfloor ~ \rfloor

—> [url=http://en.wikipedia.org/wiki/Floor_function]Floor Function.

Dissected into 54 congruent equilateral triangles

A regular hexagon, as shown in the attachment, is dissected into 54 congruent equilateral triangles by parallels to its sides. Within the figure we yield exactly 37 points which are vertices of at least one of those triangles. Those points are enumerated in an arbitrary way. A triangle is called clocky if running in a clockwise direction from the vertex with the smallest assigned number, we pass a medium number and finally reach the vertex with the highest number. Prove that at least 19 out of 54 triangles are clocky.

2007 Bundeswettbewerb Mathematik

Subcontests

Each positive integer shall be coloured red or green

Hojoo and Kestutis make moves in turn

German angle bisecting problem

Remove four points from R^3

A regular polygon with 2007 vertices

Linear and quadratic sum of digits property

Solve this equation.

Dissected into 54 congruent equilateral triangles