MathDB

2019 IGO Elementary P3

Source: 6th Iranian Geometry Olympiad (Elementary) P3

9/20/2019

There are

n>2

lines on the plane in general position; Meaning any two of them meet, but no three are concurrent. All their intersection points are marked, and then all the lines are removed, but the marked points are remained. It is not known which marked point belongs to which two lines. Is it possible to know which line belongs where, and restore them all?
Proposed by Boris Frenkin - Russia

IGOIrangeometrycombinatorics

2019 IGO Intermediate P3

Source: 6th Iranian Geometry Olympiad (Intermediate) P3

9/20/2019

Three circles

\omega_1

,

\omega_2

and

\omega_3

pass through one common point, say

P

. The tangent line to

\omega_1

at

P

intersects

\omega_2

and

\omega_3

for the second time at points

P_{1,2}

and

P_{1,3}

, respectively. Points

P_{2,1}

,

P_{2,3}

,

P_{3,1}

and

P_{3,2}

are similarly defined. Prove that the perpendicular bisector of segments

P_{1,2}P_{1,3}

,

P_{2,1}P_{2,3}

and

P_{3,1}P_{3,2}

are concurrent.
Proposed by Mahdi Etesamifard

IGOIrangeometry

2019 IGO Advanced P3

Source: 6th Iranian Geometry Olympiad (Advanced) P3

9/20/2019

Circles

\omega_1

and

\omega_2

have centres

O_1

and

O_2

, respectively. These two circles intersect at points

X

and

Y

.

AB

is common tangent line of these two circles such that

A

lies on

\omega_1

and

B

lies on

\omega_2

. Let tangents to

\omega_1

and

\omega_2

at

X

intersect

O_1O_2

at points

K

and

L

, respectively. Suppose that line

BL

intersects

\omega_2

for the second time at

M

and line

AK

intersects

\omega_1

for the second time at

N

. Prove that lines

AM, BN

and

O_1O_2

concur.
Proposed by Dominik Burek - Poland

IGOIrangeometryprojective geometrycyclic quadrilateral2019

3

Problems(3)