MathDB

1

\sum_{j=1}^p j^m is divisible by p

p

is a prime and

m

is a non-negative integer

< p-1

. Show that

\sum_{j=1}^p j^m

is divisible by

p

.

1

a square of side 1 is covered with m^2 rectangles, perimeter inequality

A square of side

1

is covered with

m^2

rectangles. Show that there is a rectangle with perimeter at least

\frac{4}{m}

.

2

1

max volume of tetrahedron which has 3 faces with area 1

Find the maximum possible volume of a tetrahedron which has three faces with area

1

.

5

1

chess tournament with 2n players, at most n^2 matches

There is a chess tournament with

2n

players (

n > 1

). There is at most one match between each pair of players. If it is not possible to find three players who all play each other, show that there are at most

n^2

matches. Conversely, show that if there are at most

n^2

matches, then it is possible to arrange them so that we cannot find three players who all play each other.

4

1

f(x) \ge f'(x) , f(x) is real polynomial of degree n

Find all

n

such that there is a real polynomial

f(x)

of degree

n

such that

f(x) \ge f'(x)

for all real

x

.

6

1

3 right angles given, 4 points concyclic wanted

ABC

is a triangle. The feet of the perpendiculars from

B

and

C

to the angle bisector at

A

are

K, L

respectively.

N

is the midpoint of

BC

, and

AM

is an altitude. Show that

K,L,N,M

are concyclic.

1986 Polish MO Finals

Subcontests

\sum_{j=1}^p j^m is divisible by p

a square of side 1 is covered with m^2 rectangles, perimeter inequality

max volume of tetrahedron which has 3 faces with area 1

chess tournament with 2n players, at most n^2 matches

f(x) \ge f'(x) , f(x) is real polynomial of degree n

3 right angles given, 4 points concyclic wanted