MathDB

2

Polygon with a center of symmetry

W

is a polygon which has a center of symmetry

S

such that if

P

belongs to

W

, then so does

P'

, where

S

is the midpoint of

PP'

. Show that there is a parallelogram

V

containing

W

such that the midpoint of each side of

V

lies on the border of

W

.

Tetrahedron inside a hemisphere of radius 1

Find the largest possible volume for a tetrahedron which lies inside a hemisphere of radius

1

.

2

permutation and expected value

For a permutation

P = (p_1, p_2, ... , p_n)

of

(1, 2, ... , n)

define

X(P)

as the number of

j

such that

p_i < p_j

for every

i < j

. What is the expected value of

X(P)

if each permutation is equally likely?

sequence a_{n+3} = a_{n+2}a_{n+1} + a_n

The sequence

a_1, a_2, a_3, ...

is defined by

a_1 = a_2 = a_3 = 1

,

a_{n+3} = a_{n+2}a_{n+1} + a_n

. Show that for any positive integer

r

we can find

s

such that

a_s

is a multiple of

r

.

1

2

inequality with reals in (0,1)

The real numbers

x_1, x_2, ... , x_n

belong to the interval

(0,1)

and satisfy

x_1 + x_2 + ... + x_n = m + r

, where

m

is an integer and

r \in [0,1)

. Show that

x_1 ^2 + x_2 ^2 + ... + x_n ^2 \leq m + r^2

.

continous function f: [0,d] to R

d

is a positive integer and

f : [0,d] \rightarrow \mathbb{R}

is a continuous function with

f(0) = f(d)

. Show that there exists

x \in [0,d-1]

such that

f(x) = f(x+1)

.

1988 Polish MO Finals

Subcontests

Polygon with a center of symmetry

Tetrahedron inside a hemisphere of radius 1

permutation and expected value

sequence a_{n+3} = a_{n+2}a_{n+1} + a_n

inequality with reals in (0,1)

continous function f: [0,d] to R