1958 Polish MO Finals

Part of Polish MO Finals

Subcontests

(6)

1

min perimeter of tangential quad

Prove that of all the quadrilaterals circuscribed around a given circle, the square has the smallest perimeter.

1

plane bisector of any dihedral angle in tetrahedron

Prove the theorem:
In a tetrahedron, the plane bisector of any dihedral angle divides the opposite edge into segments proportional to the areas of the tetrahedron faces that form this dihedral angle.

1

(1 + x)(1 + x^2) (1 + x^4) ...(1 + x^{2^k}) = 1 + x + x^2 + x^3+ .. x^m

k

is a natural number, then

(1 + x)(1 + x^2) (1 + x^4) \ldots (1 + x^{2^k}) =1 + x + x^2 + x^3+ \ldots + x^m

m

is a natural number dependent on

k

m

1

cos 2\pi / n + cos 4\pi / n + cos 6\pi/ n +...+ \cos 2n \pi/n =0

n

is a natural number greater than

1

\cos \frac{2\pi}{n} + \cos \frac{4\pi}{n} + \cos \frac{6\pi}{n} + \ldots + \cos \frac{2n \pi}{n} = 0.

1

center of gravity of quad coincide

Each side of a convex quadrilateral

ABCD

is divided into three equal parts; a straight line is drawn through the dividing points of sides

AB

AD

that lie closer to vertex

A

, and similarly for vertices

B

C

D

. Prove that the center of gravity of the quadrilateral formed by the drawn lines coincides with the center of gravity of quadrilateral

ABCD

1

504 / product of 2 consecutive swith middle = n^3

Prove that the product of three consecutive natural numbers, the middle of which is the cube of a natural number, is divisible by

504