MathDB

level 2
p5. Let

ABC

be a triangle and

M

be a variable point on

AB

N

is the point on the prolongation of

AC

such that

CN = BM

and that it does not belong to the ray

CA

. The parallelogram

BMNP

is constructed (in that order). Find the locus of

P

M

varies.
p6. Let

ABCD

be a quadrilateral. Let

C_1, C_2, C_3, C_4

be the circles of diameters

AB, BC, CD

and

DA

respectively. Let

P, Q, R

and

S

be the points of intersection (which are not vertices of

ABCD

) of

C_1

and

C_2

C_2

and

C_3

C_3

and

C_4

C_4

and

C_1

respectively. Show that the quadrilaterals

ABCD

and

PQRS

are similar.
p7.

M, N

, and P are three collinear points, with

N

between

M

and

P

. Let

r

be the perpendicualr bisector of

NP

. A point

O

is taken over

r

\omega

is the circle with center

O

passing through

N

. The tangents through

M

\omega

intersect

\omega

T

and

T'

. Find the locus of the centroid of the triangle

MTT'

O

varies over

r

.
p8.

P, Q

and

R

are the centers of three circles that pass through the same point

O

. Let

A

B

and

C

be the points (other than

O

) of intersection of the circles. Prove that

A, B

and

C

are collinear if and only if

O, Q, P

and

R

are in the same circle.
[hide=PS.] p8 might have a typo, as [url=https://www.oma.org.ar/enunciados/2da2da.htm]here in my source it was incorrect, and I tried correcting it.

Problem Statement