MathDB

2

9 lattice points in 3D, at least one midpoint is lattice point

Nine lattice points (i.e. with integer coordinates)

P_1,P_2,...,P_9

are given in space. Show that the midpoint of at least one of the segments

P_iP_j

, where

1 \le i < j \le 9

, is a lattice point as well.

Bundeswettbewerb Mathematik 1976 Problem 2.1

Prove that if

n

is an odd natural number, then

1^n +2^n +\cdots +n^n

is divisible by

n^2

.

3

2

Bundeswettbewerb Mathematik 1976 Problem 1.3

A set

S

of rational numbers is ordered in a tree-diagram in such a way that each rational number

\frac{a}{b}

(where

a

and

b

are coprime integers) has exactly two successors:

\frac{a}{a+b}

and

\frac{b}{a+b}

. How should the initial element be selected such that this tree contains the set of all rationals

r

with

0 < r < 1

? Give a procedure for determining the level of a rational number

\frac{p}{q}

in this tree.

Bundeswettbewerb Mathematik 1976 Problem 2.3

A circle is divided by

2n

points into

2n

equal arcs. Let

P_1, P_2, \ldots, P_{2n}

be an arbitrary permutation of the

2n

division points. Prove that the polygonal line

P_1 P_2 \cdots P_{2n} P_1

contains at least two parallel segments.

2

Bundeswettbewerb Mathematik 1976 Problem 1.2

Each of the two opposite sides of a convex quadrilateral is divided into seven equal parts, and corresponding division points are connected by a segment, thus dividing the quadrilateral into seven smaller quadrilaterals. Prove that the area of at least one of the small quadrilaterals equals 1\slash 7 slash of the area of the large quadrilateral.

Bundeswettbewerb Mathematik 1976 Problem 2.2

Two congruent squares

Q

and

Q'

are given in the plane. Show that they can be divided into parts

T_1, T_2, \ldots , T_n

and

T'_1 , T'_2 , \ldots , T'_n

, respectively, such that

T'_i

is the image of

T_i

under a translation for

i=1,2, \ldots, n.

4

2

at most 3n-6 segments by n distinct points under conditions

In a plane are given

n > 2

distinct points. Some pairs of these points are connected by segments so that no two of the segments intersect. Prove that there are at most

3n-6

segments.

Euclidean space

Each vertex of the 3-dimensional Euclidean space either is coloured red or blue. Prove that within those squares being possible in this space with edge length 1 there is at least one square either with three red vertices or four blue vertices !

1976 Bundeswettbewerb Mathematik

Subcontests

9 lattice points in 3D, at least one midpoint is lattice point

Bundeswettbewerb Mathematik 1976 Problem 2.1

Bundeswettbewerb Mathematik 1976 Problem 1.3

Bundeswettbewerb Mathematik 1976 Problem 2.3

Bundeswettbewerb Mathematik 1976 Problem 1.2

Bundeswettbewerb Mathematik 1976 Problem 2.2

at most 3n-6 segments by n distinct points under conditions

Euclidean space