MathDB

1

3 circles, 1 centered in the intersection of the 2 others

C_1

and

C_2

intersect in points

A

and

B

. A circle

C

with center in

A

intersect

C_1

in

M

and

P

and

C_2

in

N

and

Q

so that

N

and

Q

are located on different sides wrt

MP

and

AB> AM

. Prove that

\angle MBQ = \angle NBP

.

2

1

maximum possible number of triangles with same area

Let

E, F

be two distinct points inside a parallelogram

ABCD

. Determine the maximum possible number of triangles having the same area with three vertices from points

A, B, C, D, E, F

.

3

1

3 circles tangent internally to circumcircle and sides ABC

Let

ABC

be a triangle inscribed in circle

C

. Circles

C_1, C_2, C_3

are tangent internally with circle

C

in

A_1, B_1, C_1

and tangent to sides

[BC], [CA], [AB]

in points

A_2, B_2, C_2

respectively, so that

A, A_1

are on one side of

BC

and so on. Lines

A_1A_2, B_1B_2

and

C_1C_2

intersect the circle

C

for second time at points

A’,B’

and

C’

, respectively. If

M = BB’ \cap CC’

, prove that

m (\angle MAA’) = 90^\circ

.

4

1

4 semicircles inside a right triangle

Let

ABC

be a triangle with

m (\angle C) = 90^\circ

and the points

D \in [AC], E\in [BC]

. Inside the triangle we construct the semicircles

C_1, C_2, C_3, C_4

of diameters

[AC], [BC], [CD], [CE]

and let

\{C, K\} = C_1 \cap C_2, \{C, M\} =C_3 \cap C_4, \{C, L\} = C_2 \cap C_3, \{C, N\} =C_1 \cap C_4

. Show that points

K, L, M, N

are concyclic.

2004 JBMO Shortlist

Subcontests

3 circles, 1 centered in the intersection of the 2 others

maximum possible number of triangles with same area

3 circles tangent internally to circumcircle and sides ABC

4 semicircles inside a right triangle