MathDB

1

combinatorics with 9004 lemmings

9004

lemmings, including an equal number of both sexes, cross in rank and file the new bridge from Eyjafjallajokull to Katla. The entire column therefore moves equidistantly and at a constant speed about the bridge, whereby this is able to hold exactly half of the lemmings for the present distance. The bridge should fulfill as many lemming dreams as possible and at the same time preserve the species be opened briefly at some point in order to halve the total population. However, the law to prevent gender discrimination requires that exact half is female. Show that these sufficient claims are also can be done.
[hide=original wording]9004 Lemminge, davon gleich viele von beiden Geschlechtern, uberqueren in Reih und Glied die neue Brucke vom Eyjafjallajokull zum Katla. Die gesammte Kolonne bewegt sich also aquidistant und mit konstanter Geschwindigkeit uber die Brucke, wobei diese fur den vorliegenden Abstand genau die Halfte der Lemminge zu fassen vermag. Zur Erfullung moglichst vieler Lemmingtraume und gleichzeitiger Arterhaltung soll die Brucke irgendwann einmalig kurzzeitig aufgeklappt werden, um die Gesamtpopulation zu halbieren. Das Gesetz zur Verhinderung geschlechtsspezifischer Diskriminierung erfordert jedoch, dass davon ex akt die Halfte weiblich ist. Man zeige, dass diesen Anspruchen auch Genuge getan werden kann. https://cdn.artofproblemsolving.com/attachments/f/8/3bf1ef0f90d3eb3761ca3db04ed48480c8aab5.png

1

nxn chessboard cover only with T-tetrominos / Tetris Stones

A

T

-tetromino is a non-convex as well as non-rotationally symmetrical tetromino, which has a maximum number of outside corners (popularly also "Tetris Stone "called). Find all natural numbers

n

for which, a

n \times n

chessboard is found that can be covered only with such

T

-tetrominos.

6

1

every synogon can be broken down into a finite number of rhombuses

A synogon is a convex

2n

-gon with all sides of the same length and all opposite sides are parallel. Show that every synogon can be broken down into a finite number of rhombuses.

8

1

2player game, coloring lattice points

Albatross and Frankinfueter are playing again: each of them takes turns choosing one point in the plane with integer coordinates and paint it in his favorite color. Albatross plays first. Someone wins as soon as there is a square with all four corners in the are colored in their own color. Does anyone has a winning strategy and if so, who?

4

1

max no of bishops / horses in chessboard so that noone threatens another

How many a) bishops b) horses can be positioned on a chessboard at most, so that no one threatens another?

3

1

ever rational can be written in form of sum of numbers in form a/p^k

Show that every rational number

r

can be written as the sum of numbers in the form

\frac{a}{p^k}

where

p

is prime,

a

is an integer and

k

is natural.

2

1

n = \pm 1^2 \pm 2^2 \pm 3^2 \pm ... \pm k^2

Let

n

be an integer. Show that a natural number

k

can be found for which, the following applies with a suitable choice of signs:

n = \pm 1^2 \pm 2^2 \pm 3^2 \pm ... \pm k^2

5

1

triangle of area <=1/8 when 9 points in unit square (1973 BrMO p2)

9

points are given in the interior of the unit square. Prove there exists a triangle of area

\le \frac18

whose vertices are three of the points.

2011 QEDMO 8th

Subcontests

combinatorics with 9004 lemmings

nxn chessboard cover only with T-tetrominos / Tetris Stones

every synogon can be broken down into a finite number of rhombuses

2player game, coloring lattice points

max no of bishops / horses in chessboard so that noone threatens another

ever rational can be written in form of sum of numbers in form a/p^k

n = \pm 1^2 \pm 2^2 \pm 3^2 \pm ... \pm k^2

triangle of area <=1/8 when 9 points in unit square (1973 BrMO p2)

2011 QEDMO 8th

Subcontests

combinatorics with 9004 lemmings

nxn chessboard cover only with T-tetrominos / Tetris Stones

every synogon can be broken down into a finite number of rhombuses

2player game, coloring lattice points

max no of bishops / horses in chessboard so that noone threatens another

ever rational can be written in form of sum of numbers in form a/p^k

n = \pm 1^2 \pm 2^2 \pm 3^2 \pm ... \pm k^2

triangle of area &lt;=1/8 when 9 points in unit square (1973 BrMO p2)

triangle of area <=1/8 when 9 points in unit square (1973 BrMO p2)