2

Part of 1999 Turkey Team Selection Test

Problems(2)

Midpoints of diagonals in cyclic quadrilateral

Source: Turkey TST 1999 - P2

L

N

be the mid-points of the diagonals

[AC]

[BD]

of the cyclic quadrilateral

ABCD

, respectively. If

BD

is the bisector of the angle

ANC

, then prove that

AC

is the bisector of the angle

BLD

geometrygeometric transformationreflectionratiocircumcirclecyclic quadrilateralangle bisector

Source: Turkey TST 1999 - P5

A

B

C

D

E

F

knows a piece of gossip. They communicate by telephone via a central switchboard, which can connect only two of them at a time. During a conversation, each side tells the other everything he or she knows at that point. Determine the minimum number of calls for everyone to know all six pieces of gossip.

combinatorics proposedcombinatorics