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Indonesia Contests
Indonesia MO Shortlist
Indonesia MO Shortlist - geometry
Indonesia MO Shortlist - geometry
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Indonesia MO Shortlist
Subcontests
(18)
g12
1
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(CE-EA)/\sqrt{AB}+(AF-FB)/\sqrt{BC}+(BD-DC)/\sqrt{CA} >= ..., incircle's touches
In triangle
A
B
C
ABC
A
BC
, the incircle is tangent to
B
C
BC
BC
at
D
D
D
, to
A
C
AC
A
C
at
E
E
E
, and to
A
B
AB
A
B
at
F
F
F
. Prove that:
C
E
−
E
A
A
B
+
A
F
−
F
B
B
C
+
B
D
−
D
C
C
A
≥
B
D
−
D
C
A
B
+
C
E
−
E
A
B
C
+
A
F
−
F
B
C
A
\frac{CE-EA}{\sqrt{AB}}+\frac{AF-FB}{\sqrt{BC}} +\frac{BD-DC}{\sqrt{CA}} \ge \frac{BD-DC}{\sqrt{AB}} +\frac{CE-EA}{\sqrt{BC}} +\frac{AF-FB}{\sqrt{CA}}
A
B
CE
−
E
A
+
BC
A
F
−
FB
+
C
A
B
D
−
D
C
≥
A
B
B
D
−
D
C
+
BC
CE
−
E
A
+
C
A
A
F
−
FB
g6
1
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AP/AD >= 1 - BC^2/(AB^2 + CA^2) , altitudes related
Suppose the points
D
,
E
,
F
D, E, F
D
,
E
,
F
lie on sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
, respectively, so that
A
D
,
B
E
,
C
F
AD, BE, CF
A
D
,
BE
,
CF
are the altitudes. Also, let
A
D
AD
A
D
and
E
F
EF
EF
intersect at
P
P
P
. Prove that
A
P
A
D
≥
1
−
B
C
2
A
B
2
+
C
A
2
\frac{AP}{AD} \ge 1 - \frac{BC^2}{AB^2 + CA^2}
A
D
A
P
≥
1
−
A
B
2
+
C
A
2
B
C
2
g11
1
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CE^2/BF^2= AJ x JE / AK x KF
Given triangle
A
B
C
ABC
A
BC
and point
P
P
P
on the circumcircle of triangle
A
B
C
ABC
A
BC
. Suppose the line
C
P
CP
CP
intersects line
A
B
AB
A
B
at point
E
E
E
and line
B
P
BP
BP
intersect line
A
C
AC
A
C
at point
F
F
F
. Suppose also the perpendicular bisector of
A
B
AB
A
B
intersects
A
C
AC
A
C
at point
K
K
K
and the perpendicular bisector of
A
C
AC
A
C
intersects
A
B
AB
A
B
at point
J
J
J
. Prove that
(
C
E
B
F
)
2
=
A
J
⋅
J
E
A
K
⋅
K
F
\left( \frac{CE}{BF}\right)^2= \frac{AJ \cdot JE }{ AK \cdot KF}
(
BF
CE
)
2
=
A
K
⋅
K
F
A
J
⋅
J
E
g1
2
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collinear wanted, angle bisector, 2 parallel, 1 perpendicular
Given triangle
A
B
C
ABC
A
BC
,
A
L
AL
A
L
bisects angle
∠
B
A
C
\angle BAC
∠
B
A
C
with
L
L
L
on side
B
C
BC
BC
. Lines
L
R
LR
L
R
and
L
S
LS
L
S
are parallel to
B
A
BA
B
A
and
C
A
CA
C
A
respectively,
R
R
R
on side
A
C
AC
A
C
and
S
S
S
on side
A
B
AB
A
B
, respectively. Through point
B
B
B
draw a perpendicular on
A
L
AL
A
L
, intersecting
L
R
LR
L
R
at
M
M
M
. If point
D
D
D
is the midpoint of
B
C
BC
BC
, prove that that the three points
A
,
M
,
D
A, M, D
A
,
M
,
D
lie on a straight line.
DE, DF tangents to circle (AEF)
In triangle
A
B
C
ABC
A
BC
, let
D
D
D
be the midpoint of
B
C
BC
BC
, and
B
E
BE
BE
,
C
F
CF
CF
are the altitudes. Prove that
D
E
DE
D
E
and
D
F
DF
D
F
are both tangents to the circumcircle of triangle
A
E
F
AEF
A
EF
g10
4
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g9
4
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g8
4
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g7
4
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g5
4
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g4
3
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PQ^3 /PN^2 = PS x RS /NS wanted, starting with 2 internally tangent circles
Given that two circles
σ
1
\sigma_1
σ
1
and
σ
2
\sigma_2
σ
2
internally tangent at
N
N
N
so that
σ
2
\sigma_2
σ
2
is inside
σ
1
\sigma_1
σ
1
. The points
Q
Q
Q
and
R
R
R
lies at
σ
1
\sigma_1
σ
1
and
σ
2
\sigma_2
σ
2
, respectively, such that
N
,
R
,
Q
N,R,Q
N
,
R
,
Q
are collinear. A line through
Q
Q
Q
intersects
σ
2
\sigma_2
σ
2
at
S
S
S
and intersects
σ
1
\sigma_1
σ
1
at
O
O
O
. The line through
N
N
N
and
S
S
S
intersects
σ
1
\sigma_1
σ
1
at
P
P
P
. Prove that
P
Q
3
P
N
2
=
P
S
⋅
R
S
N
S
.
\frac{PQ^3}{PN^2} = \frac{PS \cdot RS}{NS}.
P
N
2
P
Q
3
=
NS
PS
⋅
RS
.
AP/AD >= 1 - BC/(AB + CA), touchpoints with incircle
Let
D
,
E
,
F
D, E, F
D
,
E
,
F
, be the touchpoints of the incircle in triangle
A
B
C
ABC
A
BC
with sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
, respectively, . Also, let
A
D
AD
A
D
and
E
F
EF
EF
intersect at
P
P
P
. Prove that
A
P
A
D
≥
1
−
B
C
A
B
+
C
A
\frac{AP}{AD} \ge 1 - \frac{BC}{AB + CA}
A
D
A
P
≥
1
−
A
B
+
C
A
BC
.
TA, TB,TC are sidelengths of triangle if T interior of equilateral ABC
Inside the equilateral triangle
A
B
C
ABC
A
BC
lies the point
T
T
T
. Prove that
T
A
TA
T
A
,
T
B
TB
TB
and
T
C
TC
TC
are the lengths of the sides of a triangle.
g3
4
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g2
3
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RS = \sqrt2 SQ if AD=CR , CR _|_ AB, ABC right isosceles
Let
A
B
C
ABC
A
BC
be an isosceles triangle right at
C
C
C
and
P
P
P
any point on
C
B
CB
CB
. Let also
Q
Q
Q
be the midpoint of
A
B
AB
A
B
and
R
,
S
R, S
R
,
S
be the points on
A
P
AP
A
P
such that
C
R
CR
CR
is perpendicular to
A
P
AP
A
P
and
∣
A
S
∣
=
∣
C
R
∣
|AS|=|CR|
∣
A
S
∣
=
∣
CR
∣
. Prove that the
∣
R
S
∣
=
2
∣
S
Q
∣
|RS| = \sqrt2 |SQ|
∣
RS
∣
=
2
∣
SQ
∣
.
<BZC =90^o wanted, touchpoints with incircle, _|_ bisector
Given an acute triangle
A
B
C
ABC
A
BC
. The inscribed circle of triangle
A
B
C
ABC
A
BC
is tangent to
A
B
AB
A
B
and
A
C
AC
A
C
at
X
X
X
and
Y
Y
Y
respectively. Let
C
H
CH
C
H
be the altitude. The perpendicular bisector of the segment
C
H
CH
C
H
intersects the line
X
Y
XY
X
Y
at
Z
Z
Z
. Prove that
∠
B
Z
C
=
9
0
o
.
\angle BZC = 90^o.
∠
BZC
=
9
0
o
.
intersection points of common tangents of 2 circles are concyclic
It is known that two circles have centers at
P
P
P
and
Q
Q
Q
. Prove that the intersection points of the two internal common tangents of the two circles with their two external common tangents lie on the same circle.
g4.8
1
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Indonesia National Science Olympiad 2010 - Day 2 Problem 8
Given an acute triangle
A
B
C
ABC
A
BC
with circumcenter
O
O
O
and orthocenter
H
H
H
. Let
K
K
K
be a point inside
A
B
C
ABC
A
BC
which is not
O
O
O
nor
H
H
H
. Point
L
L
L
and
M
M
M
are located outside the triangle
A
B
C
ABC
A
BC
such that
A
K
C
L
AKCL
A
K
C
L
and
A
K
B
M
AKBM
A
K
BM
are parallelogram. At last, let
B
L
BL
B
L
and
C
M
CM
CM
intersects at
N
N
N
, and let
J
J
J
be the midpoint of
H
K
HK
HK
. Show that
K
O
N
J
KONJ
K
ON
J
is also a parallelogram.Raja Oktovin, Pekanbaru
g6.2
1
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Indonesia National Science Olympiad 2010 - Day 1 Problem 2
Given an acute triangle
A
B
C
ABC
A
BC
with
A
C
>
B
C
AC>BC
A
C
>
BC
and the circumcenter of triangle
A
B
C
ABC
A
BC
is
O
O
O
. The altitude of triangle
A
B
C
ABC
A
BC
from
C
C
C
intersects
A
B
AB
A
B
and the circumcircle at
D
D
D
and
E
E
E
, respectively. A line which passed through
O
O
O
which is parallel to
A
B
AB
A
B
intersects
A
C
AC
A
C
at
F
F
F
. Show that the line
C
O
CO
CO
, the line which passed through
F
F
F
and perpendicular to
A
C
AC
A
C
, and the line which passed through
E
E
E
and parallel with
D
O
DO
D
O
are concurrent.Fajar Yuliawan, Bandung
g11.8
1
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Finding a cyclic quadrilateral
Given an acute triangle
A
B
C
ABC
A
BC
. The incircle of triangle
A
B
C
ABC
A
BC
touches
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
respectively at
D
,
E
,
F
D,E,F
D
,
E
,
F
. The angle bisector of
∠
A
\angle A
∠
A
cuts
D
E
DE
D
E
and
D
F
DF
D
F
respectively at
K
K
K
and
L
L
L
. Suppose
A
A
1
AA_1
A
A
1
is one of the altitudes of triangle
A
B
C
ABC
A
BC
, and
M
M
M
be the midpoint of
B
C
BC
BC
. (a) Prove that
B
K
BK
B
K
and
C
L
CL
C
L
are perpendicular with the angle bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
. (b) Show that
A
1
K
M
L
A_1KML
A
1
K
M
L
is a cyclic quadrilateral.
g2.3
1
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Ratios in length of a triangle
For every triangle
A
B
C
ABC
A
BC
, let
D
,
E
,
F
D,E,F
D
,
E
,
F
be a point located on segment
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
, respectively. Let
P
P
P
be the intersection of
A
D
AD
A
D
and
E
F
EF
EF
. Prove that: \frac{AB}{AF}\times DC\plus{}\frac{AC}{AE}\times DB\equal{}\frac{AD}{AP}\times BC
g6.7
2
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The sum of triangles' area
Given triangle
A
B
C
ABC
A
BC
with sidelengths
a
,
b
,
c
a,b,c
a
,
b
,
c
. Tangents to incircle of
A
B
C
ABC
A
BC
that parallel with triangle's sides form three small triangle (each small triangle has 1 vertex of
A
B
C
ABC
A
BC
). Prove that the sum of area of incircles of these three small triangles and the area of incircle of triangle
A
B
C
ABC
A
BC
is equal to \frac{\pi (a^{2}\plus{}b^{2}\plus{}c^{2})(b\plus{}c\minus{}a)(c\plus{}a\minus{}b)(a\plus{}b\minus{}c)}{(a\plus{}b\plus{}c)^{3}} (hmm,, looks familiar, isn't it? :wink: )
The same area
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram.
E
E
E
and
F
F
F
are on
B
C
,
C
D
BC, CD
BC
,
C
D
respectively such that the triangles
A
B
E
ABE
A
BE
and
B
C
F
BCF
BCF
have the same area. Let
B
D
BD
B
D
intersect
A
E
,
A
F
AE, AF
A
E
,
A
F
at
M
,
N
M, N
M
,
N
respectively. Prove there exists a triangle whose side lengths are
B
M
,
M
N
,
N
D
BM, MN, ND
BM
,
MN
,
N
D
.
g1.1
2
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Concurrent circles
Given triangle
A
B
C
ABC
A
BC
. Points
D
,
E
,
F
D,E,F
D
,
E
,
F
outside triangle
A
B
C
ABC
A
BC
are chosen such that triangles
A
B
D
ABD
A
B
D
,
B
C
E
BCE
BCE
, and
C
A
F
CAF
C
A
F
are equilateral triangles. Prove that cicumcircles of these three triangles are concurrent.
Distances on a parallelogram
A
B
C
D
ABCD
A
BC
D
is a parallelogram.
g
g
g
is a line passing
A
A
A
. Prove that the distance from
C
C
C
to
g
g
g
is either the sum or the difference of the distance from
B
B
B
to
g
g
g
, and the distance from
D
D
D
to
g
g
g
.