MathDB

concurrence

Source: Taiwan 3rd TST 2005, 3rd independent study, problem 2

8/12/2005

Given a triangle

ABC

,

A_1

divides the length of the path

CAB

into two equal parts, and define

B_1

and

C_1

analogously. Let

l_A

,

l_B

,

l_C

be the lines passing through

A_1

,

B_1

and

C_1

and being parallel to the bisectors of

\angle A

,

\angle B

, and

\angle C

. Show that

l_A

,

l_B

,

l_C

are concurrent.

geometry proposedgeometry

prime

Source: Taiwan 3rd TST 2005, 1st independent study, problem 2

8/12/2005

Find all primes

p

such that the number of distinct positive factors of

p^2+2543

is less than 16.

number theory proposednumber theory

isosceles

Source: Taiwan 2nd TST 2005, 2nd independent study, problem 2

8/12/2005

It is known that

\triangle ABC

is an acute triangle. Let

C'

be the foott of the perpendicular from

C

to

AB

, and

D

,

E

two distinct points on

CC'

. The feet of the perpendiculars from

D

to

AC

and

BC

are

F

and

G

, respectively. Show that if

DGEF

is a parallelogram then

ABC

is isosceles.

geometryparallelogramrhombusgeometry proposed

incenter

Source: Taiwan 3rd TST 2005, final exam, second day, problem 5

8/12/2005

Given a triangle

ABC

, we construct a circle

\Gamma

through

B,C

with center

O

.

\Gamma

intersects

AC, AB

at points

D

,

E

, respectively(

D

,

E

are distinct from

B

and

C

). Let the intersection of

BD

and

CE

be

F

. Extend

OF

so that it intersects the circumcircle of

\triangle ABC

at

P

. Show that the incenters of triangles

PBD

and

PCE

coincide.

geometryincentercircumcirclegeometry proposed

2

Problems(4)