1987 Bundeswettbewerb Mathematik

Part of Bundeswettbewerb Mathematik

Subcontests

(4)

2

Bundeswettbewerb Mathematik 1987 Problem 1.4

Place the integers

1,2 , \ldots, n^{3}

in the cells of a

n\times n \times n

cube such that every number appears once. For any possible enumeration, write down the maximal difference between any two adjacent cells (adjacent means having a common vertex). What is the minimal number noted down?

Bundeswettbewerb Mathematik 1987 Problem 2.4

1<k\leq n

be positive integers and

x_1 , x_2 , \ldots , x_k

be positive real numbers such that

x_1 \cdot x_2 \cdot \ldots \cdot x_k = x_1 + x_2 + \ldots +x_k.

x_{1}^{n-1} +x_{2}^{n-1} + \ldots +x_{k}^{n-1} \geq kn.

b) Find all numbers

k,n

x_1, x_2 ,\ldots , x_k

for which equality holds.

2

Bundeswettbewerb Mathematik 1987 Problem 1.2

n

be a positive integer and

M=\{1,2,\ldots, n\}.

T\subset M

is called heavy if each of its elements is greater or equal than

|T|.

f(n)

denote the number of heavy subsets of

M.

Describe a method for finding

f(n)

and use it to calculate

f(32).

Bundeswettbewerb Mathematik 1987 Problem 2.2

An arrow is assigned to each edge of a polyhedron such that for each vertex, there is an arrow pointing towards that vertex and an arrow pointing away from that vertex. Prove that there exist at least two faces such that the arrows on their boundaries form a cycle.

1

Bundeswettbewerb Mathematik 1987 Problem 1.1

p>3

n

a positive integer such that

p^n

20

digits. Prove that at least one digit appears more than twice in this number.

1

a circle covers a convex polygon

Prove that for every convex polygon, we can choose three of its consecutive vertices, such that the circle, defined by them, covers the the entire polygon.
(proposed by J. Tabov)