MathDB
Problems
Contests
National and Regional Contests
China Contests
XES Mathematics Olympiad
the 5th XMO
the 5th XMO
Part of
XES Mathematics Olympiad
Subcontests
(1)
1
1
Hide problems
2 points concyclic with (BHC)
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be an acute triangle with altitudes
A
D
AD
A
D
,
B
E
BE
BE
,
C
F
CF
CF
and orthocenter
H
H
H
. Circle
⊙
V
\odot V
⊙
V
is the circumcircle of
△
D
E
F
\vartriangle DE F
△
D
EF
. Let segments
F
D
FD
F
D
,
B
H
BH
B
H
intersect at point
P
P
P
. Let segments
E
D
ED
E
D
,
H
C
HC
H
C
intersect at point
Q
Q
Q
. Let
K
K
K
be a point on
A
C
AC
A
C
such that
V
K
⊥
C
F
VK \perp CF
V
K
⊥
CF
. a) Prove that
△
P
Q
H
∼
△
A
K
V
\vartriangle PQH \sim \vartriangle AKV
△
PQ
H
∼
△
A
K
V
. b) Let line
P
Q
PQ
PQ
intersect
⊙
V
\odot V
⊙
V
at points
I
,
G
I,G
I
,
G
. Prove that points
B
,
I
,
H
,
G
,
C
B,I,H,G,C
B
,
I
,
H
,
G
,
C
are concyclic with center the symmetric point
X
X
X
of circumcenter
O
O
O
of
△
A
B
C
\vartriangle ABC
△
A
BC
wrt
B
C
BC
BC
.[hide=PS.] There is a chance that those in the hide were not wanted in the problem, as I tried to understand the wording from a solutions' video. I couldn't find the original wording pdf or picture.https://cdn.artofproblemsolving.com/attachments/c/3/0b934c5756461ff854d38f51ef4f76d55cbd95.png[url=https://www.geogebra.org/m/cjduebke]geogebra file