MathDB

2

every positive real number is the sum of nine triplex numbers

A real number is called triplex if it has a decimal representation in which none of

0

and

3

different digit occurs. Prove that every positive real number is the sum of nine triplex numbers.

k inside a pentagon

There are

k

points in the interior of a pentagon. Together with the vertices of the pentagon they form a

(k + 5)

-element set

M

. The area of the pentagon is defined by connecting lines between the points of

M

into sub-areas in such a way that it is divided into sub-areas in such a way that no sub-areas have a point on their interior of

M

and contains exactly three points of

M

on the boundary of each part. None of the connecting lines has a point in common with any other connecting line or pentagon side, which does not belong to

M

. With such a decomposition of the pentagon, there can be an even number of connecting lines (including the pentagon sides) go out? The answer has to be justified.

2

right triangle construction given R and r

The radii of the circumcircle and the incircle of a right triangle are given. Cconstruct that triangle with compass and ruler, describe the construction and justify why it is correct.

2 player game, removing 512, 256, 128, ..., 1 numbers from {0,1,..., 1024}

Two people

A

and

B

play the following game: They take from

\{0, 1, 2, 3,..., 1024\}

alternately

512

,

256

,

128

,

64

,

32

,

16

,

8

,

4

,

2

,

1

, numbers away where

A

first removes

512

numbers,

B

removes

256

numbers etc. Two numbers

a, b

remain (

a < b

).

B

pays

A

the amount

b - a

.

A

would like to win as much as possible,

B

would like to lose as little as possible. What profit does

A

make if does every player play optimally according to their goals? The result must be justified.

1

2

soccer ball made up from black pentagons and white hexagons

The surface of a soccer ball is made up of black pentagons and white hexagons together. On the sides of each pentagon are nothing but hexagons, while on the sides of each border of hexagons alternately pentagons and hexagons. Determine from this information about the soccer ball , the number of its pentagons and its hexagons.

trianglular billard

The figure shows a triangular pool table with sides

a

,

b

and

c

. Located at point

S

on

c

a sphere - which can be assumed as a point. After kick-off, as indicated in the figure, it runs through as a result of reflections to

a, b, a, b

and

c

(in

S

) always the same track. The reflection occurs according to law of reflection. Characterize entilrely all triangles

ABC

, which allow such an orbit, and determine the locus of

S

.
https://cdn.artofproblemsolving.com/attachments/5/b/7662943e5b9ad321226e0c5f5daa3c4ac9faaa.png

4

2

distance between 2 of n + 1 points is an integer

Let

g

be a straight line and

n

a given positive integer. Prove that there are always n different points on g to choose as well as a point not lying on g in such a way that the distance between each two of these

n + 1

points is an integer.

Weird Recursion - Weird Explicit Formula

Let

f(0), f(1), f(2), \dots

be a sequence satisfying f(0) = 0 \text{and} f(n) = n - f(f(n-1)) for

n=1,2,3,\dots

. Give a formula for

f(n)

such that its value can be immediately computed using

n

without having to compute the previous terms.

1983 Bundeswettbewerb Mathematik

Subcontests

every positive real number is the sum of nine triplex numbers

k inside a pentagon

right triangle construction given R and r

2 player game, removing 512, 256, 128, ..., 1 numbers from {0,1,..., 1024}

soccer ball made up from black pentagons and white hexagons

trianglular billard

distance between 2 of n + 1 points is an integer

Weird Recursion - Weird Explicit Formula