1995 Balkan MO

Part of Balkan MO

Subcontests

(4)

1

There are exactly k squares in T having these two points

n

be a positive integer and

\mathcal S

be the set of points

(x, y)

x, y \in \{1, 2, \ldots , n\}

\mathcal T

be the set of all squares with vertices in the set

\mathcal S

a_k

k \geq 0

) the number of (unordered) pairs of points for which there are exactly

k

\mathcal T

having these two points as vertices. Prove that

a_0 = a_2 + 2a_3

1

The solutions of the equation are integers, no perfect sq.

a

b

be natural numbers with

a > b

and having the same parity. Prove that the solutions of the equation

x^2 - (a^2 - a + 1)(x - b^2 - 1) - (b^2 + 1)^2 = 0

are natural numbers, none of which is a perfect square. Albania

1

Again geometry involving two circles that intersect

\mathcal C_1(O_1, r_1)

\mathcal C_2(O_2, r_2)

r_2 > r_1

A

B

\angle O_1AO_2 = 90^\circ

O_1O_2

\mathcal C_1

C

D

\mathcal C_2

E

F

C

E

D

F

BE

\mathcal C_1

K

AC

M

BD

\mathcal C_2

L

AF

N

\frac{ r_2}{r_1} = \frac{KE}{KM} \cdot \frac{LN}{LD} .

1

Evaluate the expression

For all real numbers

x,y

x\star y = \frac{ x+y}{ 1+xy}

. Evaluate the expression

( \cdots (((2 \star 3) \star 4) \star 5) \star \cdots ) \star 1995.