1952 Poland - Second Round

Part of Poland - Second Round

Subcontests

(6)

1

plane bisects volume of tetrahedron

Prove that a plane that passes:
a) through the centers of two opposite edges of the tetrahedron and
b) through the center of one of the other edges of the tetrahedron
divides the tetrahedron into two parts of equal volumes.
Will the thesis remain true if we reject assumption (b) ?

1

vertical mast located on tower

The vertical mast located on the tower can be seen at the greatest angle from a point on the ground whose distance from the mast axis is

a

; this angle equals the given angle

\alpha

. Calculate the height of the tower and the height of the mast.

1

1/ ab +1/ bc +1/ ca =( 1/ ab + bc + ca)

Prove that if the numbers

a

b

c

satisfy the equation

\frac{1}{ab} + \frac{1}{bc} +\frac{1}{ca} = \frac{1}{ab + bc + ca},

then two of them are opposite numbers.

1

rectangle / square inscribed in rhombus

Are the following statements true?
a) if the four vertices of a rectangle lie on the four sides of a rhombus, then the sides of the rectangle are parallel to the diagonals of the rhombus;
b) if the four vertices of a square lie on the four sides of a rhombus that is not a square, then the sides of the square are parallel to the diagonals of the rhombus.

1

S = \sqrt{abcd}, area of bicentric

a

b

c

d

are the sides of a quadrilateral in which a circle can be circumscribed and a circle can be inscribed in it, then the area

S

of the quadrilateral is given by

S = \sqrt{abcd}.

1

x^3 + ax^2 + bx + c = 0

Find the necessary and sufficient conditions that the real numbers

a

b

c

should satisfy so that the equation

x^3 + ax^2 + bx + c = 0

has three real roots creating an arithmetic progression.