2

Part of 2003 Iran MO (2nd round)

Problems(2)

n house in village - Iran NMO 2003 (Second Round) - Problem2

Source:

In a village, there are

n

n>2

and all of them are not collinear. We want to generate a water resource in the village. For doing this, point

A

is better than point

B

if the sum of the distances from point

A

to the houses is less than the sum of the distances from point

B

to the houses. We call a point ideal if there doesn’t exist any better point than it. Prove that there exist at most

1

ideal point to generate the resource.

functiongeometryinequalitiesparallelogramtriangle inequalitycombinatorics proposedcombinatorics

BT+CT <= 2R - Iran NMO 2003 (Second Round) - Problem5

Source:

\angle{A}

is the least angle in

\Delta{ABC}

D

BC

from the circumcircle of

\Delta{ABC}

. The perpendicular bisectors of the segments

AB,AC

intersect the line

AD

M,N

, respectively. Point

T

is the meet point of

BM,CN

R

is the radius of the circumcircle of

\Delta{ABC}

BT+CT\leq{2R}.

geometrycircumcircletrigonometryinequalitiestrapezoidgeometry proposed