1991 Polish MO Finals

Part of Polish MO Finals

Subcontests

(3)

2

A sum which cannot be the fifth power

N=\sum\limits_{k=1}^{60}e_k k^{k^k}

e_k \in \{-1, 1\}

k

N

cannot be the fifth power of an integer.

x+y+z \leq 2 +xyz

x, y, z

are real numbers satisfying

x^2 +y^2 +z^2 = 2

, prove the inequality

x + y + z \leq 2 + xyz

When does equality occur?

2

Lattice points and path

X

be the set of all lattice points in the plane (points

(x, y)

x, y \in \mathbb{Z}

). A path of length

n

(P_0, P_1, ... , P_n)

X

P_{i-1}P_i = 1

i = 1, ... , n

F(n)

be the number of distinct paths beginning in

P_0=(0,0)

and ending in any point

P_n

y = 0

F(n) = \binom{2n}{n}

Two noncongruent circles

Two noncongruent circles

k_1

k_2

are exterior to each other. Their common tangents intersect the line through their centers at points

A

B

P

be any point of

k_1

. Prove that there is a diameter of

k_2

with one endpoint on line

PA

and the other on

PB

2

Two tetrahedra

Prove or disprove that there exist two tetrahedra

T_1

T_2

such that: (i) the volume of

T_1

is greater than that of

T_2

; (ii) the area of any face of

T_1

does not exceed the area of any face of

T_2

Functions operating on vectors

On the Cartesian plane consider the set

V

of all vectors with integer coordinates. Determine all functions

f : V \rightarrow \mathbb{R}

satisfying the conditions: (i)

f(v) = 1

for each of the four vectors

v \in V

of unit length. (ii)

f(v+w) = f(v)+f(w)

for every two perpendicular vectors

v, w \in V

(Zero vector is considered to be perpendicular to every vector).