1997 Balkan MO

Part of Balkan MO

Subcontests

(3)

1

Two circles, externally tangent and a third circle comes in

\mathcal C_1

\mathcal C_2

touch each other externally at

D

, and touch a circle

\omega

B

C

, respectively. Let

A

be an intersection point of

\omega

and the common tangent to

\mathcal C_1

\mathcal C_2

D

AB

AC

\mathcal C_1

\mathcal C_2

K

L

, respectively, and the line

BC

\mathcal C_1

M

\mathcal C_2

N

. Prove that the lines

AD

KM

LN

are concurrent. Greece

1

A collection of subsets each has exactly one of two elements

S = \{A_1,A_2,\ldots ,A_k\}

be a collection of subsets of an

n

A

. If for any two elements

x, y \in A

there is a subset

A_i \in S

containing exactly one of the two elements

x

y

2^k\geq n

1

Convex quadrilateral and O inside becomes square and center

O

is a point inside a convex quadrilateral

ABCD

OA^2 + OB^2 + OC^2 + OD^2 = 2\mathcal A[ABCD] ,

\mathcal A[ABCD]

we have denoted the area of

ABCD

ABCD

is a square and

O

is its center. Yugoslavia