MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2024 Vietnam National Olympiad
3
3
Part of
2024 Vietnam National Olympiad
Problems
(1)
Geometry but... lots of points!
Source: 2024 Vietnam National Olympiad - Problem 3
1/5/2024
Let
A
B
C
ABC
A
BC
be an acute triangle with circumcenter
O
O
O
. Let
A
′
A'
A
′
be the center of the circle passing through
C
C
C
and tangent to
A
B
AB
A
B
at
A
A
A
, let
B
′
B'
B
′
be the center of the circle passing through
A
A
A
and tangent to
B
C
BC
BC
at
B
B
B
, let
C
′
C'
C
′
be the center of the circle passing through
B
B
B
and tangent to
C
A
CA
C
A
at
C
C
C
.a) Prove that the area of triangle
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
is not less than the area of triangle
A
B
C
ABC
A
BC
. b) Let
X
,
Y
,
Z
X, Y, Z
X
,
Y
,
Z
be the projections of
O
O
O
onto lines
A
′
B
′
,
B
′
C
′
,
C
′
A
′
A'B', B'C', C'A'
A
′
B
′
,
B
′
C
′
,
C
′
A
′
. Given that the circumcircle of triangle
X
Y
Z
XYZ
X
Y
Z
intersects lines
A
′
B
′
,
B
′
C
′
,
C
′
A
′
A'B', B'C', C'A'
A
′
B
′
,
B
′
C
′
,
C
′
A
′
again at
X
′
,
Y
′
,
Z
′
X', Y', Z'
X
′
,
Y
′
,
Z
′
(
X
′
≠
X
,
Y
′
≠
Y
,
Z
′
≠
Z
X' \neq X, Y' \neq Y, Z' \neq Z
X
′
=
X
,
Y
′
=
Y
,
Z
′
=
Z
), prove that lines
A
X
′
,
B
Y
′
,
C
Z
′
AX', BY', CZ'
A
X
′
,
B
Y
′
,
C
Z
′
are concurrent.
geometry