1995 Belarus Team Selection Test

Part of Belarus Team Selection Test

Subcontests

(3)

2

Infine Sequence

Show that there is no infinite sequence an of natural numbers such that

a_{a_n}=a_{n+1}a_{n-1}-a_{n}^2

n\geq 2

Inequality

0<a,b<1

p,q\geq 0 ,\ p+q=1

are real numbers , then prove that:

a^pb^q+(1-a)^p(1-b)^q\le 1

2

Geometry

S,S_1,S_2

are given in a plane.

S_1

S_2

touch each other externally, and both touch

S

A_1

A_2

respectively. The common internal tangent to

S_1

S_2

S

P

Q.

B_1

B_2

be the intersections of

PA_1

PA_2

S_1

S_2

, respectively. Prove that

B_1B_2

is a common tangent to

S_1,S_2

Combinatorics

There is a room having a form of right-angled parallelepiped. Four maps of the same scale are hung (generally, on different levels over the floor) on four walls of the room, so that sides of the maps are parallel to sides of the wall. It is known that the four points corresponding to each of Stockholm, Moscow, and Istanbul are coplanar. Prove that the four points coresponding to Hong Kong are coplanar as well.

2

There is a 100 &times; 100 square table

There is a 100 x100 square table, a real number being written in each cell.

A

B

play the following game. They choose, turn by turn, some row of the table (if it has not been chosen before). When

A

B

50

rows chosen each, they sum the numbers in the corresponding cells of the chosen rows, and then sum the squares of all

100

obtained numbers and compare the results.

A

player who has the greater result wins. Player

A

begins. Show that

A

can avoid a defeat.

Polynomial with odd coefficients

Prove that the number of odd coefficients in the polynomial

(1+x)^n

2

for every positive integer

N