MathDB

1

1025 cities are connected to each other by 10 airlines

P_1,...,P_{1025}

are connected to each other by airlines

A_1,...,A_{10}

so that for any two distinct cities

P_k

and

P_m

there is an airline offering a direct flight between them. Prove that one of the airlines can offer a round trip with an odd number of flights.

5

1

a regular hexagon of side $1$ is covered by six unit disks

A regular hexagon of side

1

is covered by six unit disks. Prove that none of the vertices of the hexagon is covered by two (or more) discs.

4

1

probability sequence of sums of heads contains n in heads or tails

A coin is tossed

n

times, and the outcome is written in the form (

a_1,a_2,...,a_n

), where

a_i = 1

or

2

depending on whether the result of the

i

-th toss is the head or the tail, respectively. Set

b_j = a_1 +a_2 +...+a_j

for

j = 1,2,...,n

, and let

p(n)

be the probability that the sequence

b_1,b_2,...,b_n

contains the number

n

. Express

p(n)

in terms of

p(n-1)

and

p(n-2)

.

3

1

geometri inequaliy with circles and a regular octahedron

Let

W

be a regular octahedron and

O

be its center. In a plane

P

containing

O

circles

k_1(O,r_1)

and

k_2(O,r_2)

are chosen so that

k_1 \subset P\cap W \subset k_2

. Prove that

\frac{r_1}{r_2}\le \frac{\sqrt3}{2}

2

1

\sum_{j=1}^{n}|\sum_{k=1}^{n} a_{j,p}(k)| \ge \frac{n}{2}

Let

n

be a positive integer. For all

i, j \in \{1,2,...,n\}

define

a_{j,i} = 1

if

j = i

and

a_{j,i} = 0

otherwise. Also, for

i = n+1,...,2n

and

j = 1,...,n

define

a_{j,i} = -\frac{1}{n}

. Prove that for any permutation

p

of the set

\{1,2,...,2n\}

the following inequality holds:

\sum_{j=1}^{n}\left|\sum_{k=1}^{n} a_{j,p}(k)\right| \ge \frac{n}{2}

1

real functions which map the sum of n elements into the sum of their image

Find the number of all real functions

f

which map the sum of

n

elements into the sum of their images, such that

f^{n-1}

is a constant function and

f^{n-2}

is not. Here

f^0(x) = x

and

f^k = f \circ f^{k-1}

for

k \ge 1

.

1984 Polish MO Finals

Subcontests

1025 cities are connected to each other by 10 airlines

a regular hexagon of side $1$ is covered by six unit disks

probability sequence of sums of heads contains n in heads or tails

geometri inequaliy with circles and a regular octahedron

\sum_{j=1}^{n}|\sum_{k=1}^{n} a_{j,p}(k)| \ge \frac{n}{2}

real functions which map the sum of n elements into the sum of their image