MathDB

1

tangential quad property

Prove that if a convex quadrilateral has the property that there exists a circle tangent to its sides (i.e. an inscribed circle), and also a circle tangent to the extensions of its sides (an excircle), then the diagonals of the quadrilateral are perpendicular to each other.

5

1

bisect segment

Given a segment

AB

and a line

m

parallel to this segment. Find the midpoint of the segment

AB

using only a ruler, i.e. drawing only straight lines.

4

1

\frac{a}{b + c} + \frac{b}{c+ a} + \frac{c}{a+b} \geq \frac{3}{2}

Prove that if

a > 0

,

b > 0

,

c > 0

, then

\frac{a}{b + c} + \frac{b}{c+ a} + \frac{c}{a+b} \geq \frac{3}{2}.

3

1

distnace of 2 points, cube related

Given a cube with edge

AB = a

cm. Point

M

of segment

AB

is distant from the diagonal of the cube, which is oblique to

AB

, by

k

cm. Find the distance of point

M

from the midpoint

S

of segment

AB

.

2

1

ratio of perimeters = ratio of circles

Prove that if

M

,

N

,

P

are the feet of the altitudes of acute-angled triangle

ABC

, then the ratio of the perimeter of triangle

MNP

to the perimeter of triangle

ABC

is equal to the ratio of the radius of the circle inscribed in triangle

ABC

to the radius of the circle circumscribed about triangle

ABC

.

1

\frac{n^5}{120} - \frac{n^3}{24} + \frac{n}{30} is integer

Prove that if

n

is an integer, then

\frac{n^5}{120} - \frac{n^3}{24} + \frac{n}{30}

is also an integer.

1957 Poland - Second Round

Subcontests

tangential quad property

bisect segment

\frac{a}{b + c} + \frac{b}{c+ a} + \frac{c}{a+b} \geq \frac{3}{2}

distnace of 2 points, cube related

ratio of perimeters = ratio of circles

\frac{n^5}{120} - \frac{n^3}{24} + \frac{n}{30} is integer