1994 Polish MO Finals

Part of Polish MO Finals

Subcontests

(3)

2

number of maps and limit

k

is a fixed positive integer. Let

a_n

be the number of maps

f

from the subsets of

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

\{1, 2, ... , k\}

such that for all subsets

A, B

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

f(A \cap B) = \min (f(A), f(B))

\lim_{n \to \infty} \sqrt[n]{a_n}

n distinct reals and a,b,c,d

The distinct reals

x_1, x_2, ... , x_n

n > 3

\sum_{i=1}^n x_i = 0

\sum_{i=1}^n x_i ^2 = 1

. Show that four of the numbers

a, b, c, d

a + b + c + nabc \leq \sum_{i=1}^n x_i ^3 \leq a + b + d + nabd

2

Common point as a varies

Let be given two parallel lines

k

l

, and a circle not intersecting

k

. Consider a variable point

A

k

. The two tangents from this point

A

to the circle intersect the line

l

B

C

m

be the line through the point

A

and the midpoint of the segment

BC

. Prove that all the lines

m

A

varies) have a common point.

Parallelopiped and inequality

A parallelopiped has vertices

A_1, A_2, ... , A_8

O

4 \sum_{i=1}^8 OA_i ^2 \leq \left(\sum_{i=1}^8 OA_i \right) ^2

2

triples (x,y,z) in Q+

Find all triples

(x,y,z)

of positive rationals such that

x + y + z

\dfrac{1}{x} + \dfrac{1}{y} + \dfrac{1}{z}

xyz

are all integers.

jugs with water

m, n

are relatively prime. We have three jugs which contain

m

n

m+n

liters. Initially the largest jug is full of water. Show that for any

k

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

we can get exactly

k

liters into one of the jugs.