MathDB

1

3 equal angles via projections on isosceles triangle

ABC

be an isosceles triangle

(AB=AC)

so that

\angle A< 2 \angle B

. Let

D,Z

points on the extension of height

AM

so that

\angle CBD = \angle A

and

\angle ZBA = 90^\circ

. Let

E

the orthogonal projection of

M

on height

BF

, and let

K

the orthogonal projection of

Z

on

AE

. Prove that

\angle KDZ = \angle KDB = \angle KZB

.

6

1

circumcenter lies on circumcircle, intersecting circles

Let

C_1,C_2

be two circles intersecting at points

A,P

with centers

O,K

respectively. Let

B,C

be the symmetric of

A

wrt

O,K

in circles

C_1,C_2

respectively. A random line passing through

A

intersects circles

C_1,C_2

at

D,E

respectively. Prove that the center of circumcircle of triangle

DEP

lies on the circumcircle of triangle

OKP

.

5

1

maximum length in a equilateral, on 2 concentric circles

Let

O

be the center of the concentric circles

C_1,C_2

of radii

3

and

5

respectively. Let

A\in C_1, B\in C_2

and

C

point so that triangle

ABC

is equilateral. Find the maximum length of

[OC]

.

2

1

isosceles trapezoid with 3 equal sides

Let

ABCD

be an isosceles trapezoid with

AB=AD=BC, AB//CD, AB>CD

. Let

E= AC \cap BD

and

N

symmetric to

B

wrt

AC

. Prove that the quadrilateral

ANDE

is cyclic.

7

1

6 points in a parallelogram, two equal segments

Let

ABCD

be a parallelogram.

P \in (CD), Q \in (AB)

,

M= AP \cap DQ

,

N=BP \cap CQ

,

K=MN \cap AD

,

L= MN \cap BC

. Prove that

BL=DK

.

3

1

Regular hexagon

Let

ABCDEF

be a regular hexagon and

M\in (DE)

,

N\in(CD)

such that

m (\widehat {AMN}) = 90^\circ

and

AN = CM \sqrt {2}

. Find the value of

\frac{DM}{ME}

.

2005 JBMO Shortlist

Subcontests

3 equal angles via projections on isosceles triangle

circumcenter lies on circumcircle, intersecting circles

maximum length in a equilateral, on 2 concentric circles

isosceles trapezoid with 3 equal sides

6 points in a parallelogram, two equal segments

Regular hexagon