1973 Polish MO Finals

Part of Polish MO Finals

Subcontests

(6)

1

parallelepiped wanted

W

has the following properties:
(i) It possesses a center of symmetry. (ii) The section of

W

by a plane passing through the center of symmetry and one of its edges is always a parallelogram. (iii) There is a vertex of

W

at which exactly three edges meet.
Prove that

W

is a parallelepiped.

1

n collinear points, translated

A set of segments with the total length less than

1

is given on a line. Prove that every set of

n

points on the line can be translated by a vector of length not exceeding

n/2

, so that all the obtained points are away from the given segments.

1

m/n = a sum of reciprocals of distinct positive integers

Prove that every positive rational number

m/n

can be represented as a sum of reciprocals of distinct positive integers.

1

one ellipse containing the centrally symmetric polygon

Prove that for every centrally symmetric polygon there is at most one ellipse containing the polygon and having the minimal area.

1

n n tosses, a fair coin shows the head up 100 consecutive times

p_n

denote the probability that, in

n

tosses, a fair coin shows the head up

100

consecutive times. Prove that the sequence

(p_n)

converges and determine its limit.

1

every polynomial is a difference of two increasing polynomials

Prove that every polynomial is a difference of two increasing polynomials.