1971 Bundeswettbewerb Mathematik

Part of Bundeswettbewerb Mathematik

Subcontests

(4)

2

Funny movement, but no tour possible

P

Q

be two horizontal neighbouring squares on a

n \times n

P

on the left and

Q

on the right. On the left square

P

there is a stone that shall be moved around the board. The following moves are allowed: 1) move it one square upwards 2) move it one square to the right 3) move it one square down and one square to the left (diagonal movement) Example: you can get from

e5

f5

e6

d4

. Show that for no

n

there is tour visting every square exactly once and ending in

Q

Parallel intersecting broken line 501 times

Inside a square with side lengths

1

a broken line of length

>1000

without selfintersection is drawn. Show that there is a line parallel to a side of the square that intersects the broken line in at least

501

2

Cutting paper into 8 or 12 pieces

You are given a piece of paper. You can cut the paper into

8

12

pieces. Then you can do so for any of the new pieces or let them uncut and so on. Can you get exactly

60

pieces¿ Show that you can get every number of pieces greater than

60

Error correcting codes: differing at three positions

The inhabitants of a planet speak a language only using the letters

A

O

. To avoid mistakes, any two words of equal length differ at least on three positions. Show that there are not more than

\frac{2^n}{n+1}

n

2

Acute triangle

Given five segments such that any three of them can be used to form a triangle. Show that at least one of these triangles is acute-angled. Alternative formulation: Five segments have lengths such that any three of them can be sides of a triangle. Prove that there exists at least one acute-angled triangle among these triangles.

One-way roads between cities

Between any two cities of a country there is only one one-way road. Show that there is a city from that every other city can be reached directly or by going over only one intermediate city.
I'm sure it was posted before but couldn't find it.

2

Clearing numbers and writing their difference

1,2,...,1970

are written on a board. One is allowed to remove

2

numbers and to write down their difference instead. When repeated often enough, only one number remains. Show that this number is odd.

Never a prime

For any positive integers

a,b,c,d,n

, it is given that

n

is composite, such that

n=ab=cd

S

S=a^2+b^2+c^2+d^2

S

is never a prime number