MathDB

2

Bundeswettbewerb Mathematik 1980 Problem 1.4

a_1, a_2, a_3, \ldots

with

a_n = \frac{1}{n(n+1)}.

In how many ways can the number

\frac{1}{1980}

be represented as the sum of finitely many consecutive terms of this sequence?

Bundeswettbewerb Mathematik 1980 Problem 2.4

A sequence of integers

a_1,a_2,\ldots

is defined by

a_1=1,a_2=2

and for

n\geq 1

,

a_{n+2}=\left\{\begin{array}{cl}5a_{n+1}-3a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is even},\\ a_{n+1}-a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is odd},\end{array}\right.

(a) Prove that the sequence contains infinitely many positive terms and infinitely many negative terms.
(b) Prove that no term of the sequence is zero.
(c) Show that if

n = 2^k - 1

for

k\geq 2

, then

a_n

is divisible by

7

.

2

Bundeswettbewerb Mathematik 1980 Problem 1.2

In a triangle

ABC

, the bisectors of angles

A

and

B

meet the opposite sides of the triangle at points

D

and

E

, respectively. A point

P

is arbitrarily chosen on the line

DE

. Prove that the distance of

P

from line

AB

equals the sum or the difference of the distances of

P

from lines

AC

and

BC

.

NT PHP in subset of N

Prove that from every set of

n+1

natural numbers, whose prime factors are in a given set of

n

prime numbers, one can select several distinct numbers whose product is a perfect square.

1

2

Bundeswettbewerb Mathematik 1980 Problem 1.1

Six free cells are given in a row. Players

A

and

B

alternately write digits from

0

to

9

in empty cells, with

A

starting. When all the cells are filled, one considers the obtained six-digit number

z

. Player

B

wins if

z

is divisible by a given natural number

n

, and loses otherwise. For which values of

n

not exceeding

20

can

B

win independently of his opponent’s moves?

if sum \sqrt[3]{a}+\sqrt[3]{b} is rational then a,b are perfect cubes

Let

a

and

b

be integers. Prove that if

\sqrt[3]{a}+\sqrt[3]{b}

is a rational number, then both

a

and

b

are perfect cubes.

3

2

2n+3 points in the plane

Given 2n+3 points in the plane, no three on a line and no four on a circle, prove that it is always possible to find a circle C that goes through three of the given points and splits the other 2n in half, that is, has n on the inside and n on the outside.

Bundeswettbewerb Mathematik 1980 Problem 2.3

In a triangle

ABC

, points

P, Q

and

R

distinct from the vertices of the triangle are chosen on sides

AB, BC

and

CA

, respectively. The circumcircles of the triangles

APR

,

BPQ

, and

CQR

are drawn. Prove that the centers of these circles are the vertices of a triangle similar to triangle

ABC

.

1980 Bundeswettbewerb Mathematik

Subcontests

Bundeswettbewerb Mathematik 1980 Problem 1.4

Bundeswettbewerb Mathematik 1980 Problem 2.4

Bundeswettbewerb Mathematik 1980 Problem 1.2

NT PHP in subset of N

Bundeswettbewerb Mathematik 1980 Problem 1.1

if sum \sqrt[3]{a}+\sqrt[3]{b} is rational then a,b are perfect cubes

2n+3 points in the plane

Bundeswettbewerb Mathematik 1980 Problem 2.3