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National and Regional Contests
Indonesia Contests
Indonesia Regional
Indonesia Regional MO OSP SMA - geometry
Indonesia Regional MO OSP SMA - geometry
Part of
Indonesia Regional
Subcontests
(24)
2020.1
1
Hide problems
area of rectangle, 5 equal squares (2020 Indonedia MO Province P2 q1 OSP)
In the figure, point
P
,
Q
,
R
,
S
P, Q,R,S
P
,
Q
,
R
,
S
lies on the side of the rectangle
A
B
C
D
ABCD
A
BC
D
. https://1.bp.blogspot.com/-Ff9rMibTuHA/X9PRPbGVy-I/AAAAAAAAMzA/2ytG0aqe-k0fPL3hbSp_zHrMYAfU-1Y_ACLcBGAsYHQ/s426/2020%2BIndonedia%2BMO%2BProvince%2BP2%2Bq1.png If it is known that the area of the small square is
1
1
1
unit, determine the area of the rectangle
A
B
C
D
ABCD
A
BC
D
.
2020.4
1
Hide problems
collinear wanted, perpendiculars and altitudes related
It is known that triangle
A
B
C
ABC
A
BC
is not isosceles with altitudes of
A
A
1
,
B
B
1
AA_1, BB_1
A
A
1
,
B
B
1
, and
C
C
1
CC_1
C
C
1
. Suppose
B
A
B_A
B
A
and
C
A
C_A
C
A
respectively points on
B
B
1
BB_1
B
B
1
and
C
C
1
CC_1
C
C
1
so that
A
1
B
A
A_1B_A
A
1
B
A
is perpendicular on
B
B
1
BB_1
B
B
1
and
A
1
C
A
A_1C_A
A
1
C
A
is perpendicular on
C
C
1
CC_1
C
C
1
. Lines
B
A
C
A
B_AC_A
B
A
C
A
and
B
C
BC
BC
intersect at the point
T
A
T_A
T
A
. Define in the same way the points
T
B
T_B
T
B
and
T
C
T_C
T
C
. Prove that points
T
A
,
T
B
T_A, T_B
T
A
,
T
B
, and
T
C
T_C
T
C
are collinear.
2005.1
1
Hide problems
min of max sidelength of an inscribed quadrilateral in circle of radius 1
The length of the largest side of the cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
is
a
a
a
, while the radius of the circumcircle of
△
A
C
D
\vartriangle ACD
△
A
C
D
is
1
1
1
. Find the smallest possible value for
a
a
a
. Which cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
gives the value
a
a
a
equal to the smallest value?
2003.3
1
Hide problems
area of quadril. wanted on cube (2003 Indonedia MO Province P2 q3 OSP)
The points
P
P
P
and
Q
Q
Q
are the midpoints of the edges
A
E
AE
A
E
and
C
G
CG
CG
on the cube
A
B
C
D
.
E
F
G
H
ABCD.EFGH
A
BC
D
.
EFG
H
respectively. If the length of the cube edges is
1
1
1
unit, determine the area of the quadrilateral
D
P
F
Q
DPFQ
D
PFQ
.
2007.1
1
Hide problems
ABCD with equal sides (2007 Indonedia MO Province P2 q1 OSP)
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral with
A
B
=
B
C
=
C
D
=
D
A
AB = BC = CD = DA
A
B
=
BC
=
C
D
=
D
A
. (a) Prove that point A must be outside of triangle
B
C
D
BCD
BC
D
. (b) Prove that each pair of opposite sides on
A
B
C
D
ABCD
A
BC
D
is always parallel.
2009.3
1
Hide problems
r_1 + r_2> r, inradii
Given triangle
A
B
C
ABC
A
BC
and point
D
D
D
on the
A
C
AC
A
C
side. Let
r
1
,
r
2
r_1, r_2
r
1
,
r
2
and
r
r
r
denote the radii of the incircle of the triangles
A
B
D
,
B
C
D
ABD, BCD
A
B
D
,
BC
D
, and
A
B
C
ABC
A
BC
, respectively. Prove that
r
1
+
r
2
>
r
r_1 + r_2> r
r
1
+
r
2
>
r
.
2007.4
1
Hide problems
DE + DF <= BC , 3 altitudes
In acute triangles
A
B
C
ABC
A
BC
,
A
D
,
B
E
,
C
F
AD, BE ,CF
A
D
,
BE
,
CF
are altitudes, with
D
,
E
,
F
D, E, F
D
,
E
,
F
on the sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
, respectively. Prove that
D
E
+
D
F
≤
B
C
DE + DF \le BC
D
E
+
D
F
≤
BC
2006.1
1
Hide problems
perpendicular inside a right triangle, altitude and midpoints
Suppose triangle
A
B
C
ABC
A
BC
is right-angled at
B
B
B
. The altitude from
B
B
B
intersects the side
A
C
AC
A
C
at point
D
D
D
. If points
E
E
E
and
F
F
F
are the midpoints of
B
D
BD
B
D
and
C
D
CD
C
D
, prove that
A
E
⊥
B
F
AE \perp BF
A
E
⊥
BF
.
2005.4
1
Hide problems
right triangle with integer sides and equal values of perimeter and area
The lengths of the three sides
a
,
b
,
c
a, b, c
a
,
b
,
c
with
a
≤
b
≤
c
a \le b \le c
a
≤
b
≤
c
, of a right triangle is an integer. Find all the sequences
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
so that the values of perimeter and area of the triangle are the same.
2004.5
1
Hide problems
in every 5 lattice points there is pair of lattice points that contains a third
The lattice point on the plane is a point that has coordinates in the form of a pair of integers. Let
P
1
,
P
2
,
P
3
,
P
4
,
P
5
P_1, P_2, P_3, P_4, P_5
P
1
,
P
2
,
P
3
,
P
4
,
P
5
be five different lattice points on the plane. Prove that there is a pair of points
(
P
i
,
P
j
)
,
i
≠
j
(P_i, P_j), i \ne j
(
P
i
,
P
j
)
,
i
=
j
, so that the line segment
P
i
P
j
P_iP_j
P
i
P
j
contains a lattice point other than
P
i
P_i
P
i
and
P
j
P_j
P
j
.
2004.2
1
Hide problems
AO/AD + BO/BE + CO / CF=2 wanted, cevians related
Triangle
A
B
C
ABC
A
BC
is given. The points
D
,
E
D, E
D
,
E
, and
F
F
F
are located on the sides
B
C
,
C
A
BC, CA
BC
,
C
A
and
A
B
AB
A
B
respectively so that the lines
A
D
,
B
E
AD, BE
A
D
,
BE
and
C
F
CF
CF
intersect at point
O
O
O
. Prove that
A
O
A
D
+
B
O
B
E
+
C
O
C
F
=
2
\frac{AO}{AD} + \frac{BO}{BE} + \frac{CO}{ CF}=2
A
D
A
O
+
BE
BO
+
CF
CO
=
2
2011.4
1
Hide problems
EO _|_ ZD wanted, rectangle, equilateral, equal extensions related
Given a rectangle
A
B
C
D
ABCD
A
BC
D
with
A
B
=
a
AB = a
A
B
=
a
and
B
C
=
b
BC = b
BC
=
b
. Point
O
O
O
is the intersection of the two diagonals. Extend the side of the
B
A
BA
B
A
so that
A
E
=
A
O
AE = AO
A
E
=
A
O
, also extend the diagonal of
B
D
BD
B
D
so that
B
Z
=
B
O
.
BZ = BO.
BZ
=
BO
.
Assume that triangle
E
Z
C
EZC
EZC
is equilateral. Prove that (i)
b
=
a
3
b = a\sqrt3
b
=
a
3
(ii)
E
O
EO
EO
is perpendicular to
Z
D
ZD
Z
D
2010.1
1
Hide problems
collinear wanted, AR/RB =BP/PC=CQ/QA=CP_1/P_1B, centroid
Given triangle
A
B
C
ABC
A
BC
. Suppose
P
P
P
and
P
1
P_1
P
1
are points on
B
C
,
Q
BC, Q
BC
,
Q
lies on
C
A
,
R
CA, R
C
A
,
R
lies on
A
B
AB
A
B
, such that
A
R
R
B
=
B
P
P
C
=
C
Q
Q
A
=
C
P
1
P
1
B
\frac{AR}{RB}=\frac{BP}{PC}=\frac{CQ}{QA}=\frac{CP_1}{P_1B}
RB
A
R
=
PC
BP
=
Q
A
CQ
=
P
1
B
C
P
1
Let
G
G
G
be the centroid of triangle
A
B
C
ABC
A
BC
and
K
=
A
P
1
∩
R
Q
K = AP_1 \cap RQ
K
=
A
P
1
∩
RQ
. Prove that points
P
,
G
P,G
P
,
G
, and
K
K
K
are collinear.
2012.4
1
Hide problems
AB + AC > = BC cos (<BAC) + 2AH sin(<BAC) , where AH altitude
Given an acute triangle
A
B
C
ABC
A
BC
. Point
H
H
H
denotes the foot of the altitude drawn from
A
A
A
. Prove that
A
B
+
A
C
≥
B
C
c
o
s
∠
B
A
C
+
2
A
H
s
i
n
∠
B
A
C
AB + AC \ge BC cos \angle BAC + 2AH sin \angle BAC
A
B
+
A
C
≥
BC
cos
∠
B
A
C
+
2
A
Hs
in
∠
B
A
C
2015.3
1
Hide problems
<BAC = 2< DPC wanted, BD = 2DC and <BAC = <BPD given
Given the isosceles triangle
A
B
C
ABC
A
BC
, where
A
B
=
A
C
AB = AC
A
B
=
A
C
. Let
D
D
D
be a point in the segment
B
C
BC
BC
so that
B
D
=
2
D
C
BD = 2DC
B
D
=
2
D
C
. Suppose also that point
P
P
P
lies on the segment
A
D
AD
A
D
such that:
∠
B
A
C
=
∠
B
P
D
\angle BAC = \angle BP D
∠
B
A
C
=
∠
BP
D
. Prove that
∠
B
A
C
=
2
∠
D
P
C
\angle BAC = 2\angle DP C
∠
B
A
C
=
2∠
D
PC
.
2013.5
1
Hide problems
< ABC <= 60^o if longest altitude AD=median BE
Given an acute triangle
A
B
C
ABC
A
BC
. The longest line of altitude is the one from vertex
A
A
A
perpendicular to
B
C
BC
BC
, and it's length is equal to the length of the median of vertex
B
B
B
. Prove that
∠
A
B
C
≤
6
0
o
\angle ABC \le 60^o
∠
A
BC
≤
6
0
o
2014.4
1
Hide problems
AN x AE = AD x AF = AB x AC , 2 circles related
Let
Γ
\Gamma
Γ
be the circumcircle of triangle
A
B
C
ABC
A
BC
. One circle
ω
\omega
ω
is tangent to
Γ
\Gamma
Γ
at
A
A
A
and tangent to
B
C
BC
BC
at
N
N
N
. Suppose that the extension of
A
N
AN
A
N
crosses
Γ
\Gamma
Γ
again at
E
E
E
. Let
A
D
AD
A
D
and
A
F
AF
A
F
be respectively the line of altitude
A
B
C
ABC
A
BC
and diameter of
Γ
\Gamma
Γ
, show that
A
N
×
A
E
=
A
D
×
A
F
=
A
B
×
A
C
AN \times AE = AD \times AF = AB \times AC
A
N
×
A
E
=
A
D
×
A
F
=
A
B
×
A
C
2014.2
1
Hide problems
angle bisector wanted, 2 excenters related
Given an acute triangle
A
B
C
ABC
A
BC
with
A
B
<
A
C
AB <AC
A
B
<
A
C
. The ex-circles of triangle
A
B
C
ABC
A
BC
opposite
B
B
B
and
C
C
C
are centered on
B
1
B_1
B
1
and
C
1
C_1
C
1
, respectively. Let
D
D
D
be the midpoint of
B
1
C
1
B_1C_1
B
1
C
1
. Suppose that
E
E
E
is the point of intersection of
A
B
AB
A
B
and
C
D
CD
C
D
, and
F
F
F
is the point of intersection of
A
C
AC
A
C
and
B
D
BD
B
D
. If
E
F
EF
EF
intersects
B
C
BC
BC
at point
G
G
G
, prove that
A
G
AG
A
G
is the bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
.
2017.3
1
Hide problems
find point X on BC such that symmetric of orthocenter wrt lies on circumcircle
Given triangle
A
B
C
ABC
A
BC
, the three altitudes intersect at point
H
H
H
. Determine all points
X
X
X
on the side
B
C
BC
BC
so that the symmetric of
H
H
H
wrt point
X
X
X
lies on the circumcircle of triangle
A
B
C
ABC
A
BC
.
2018.3
1
Hide problems
parallel wanted, starting with equal tangent circles
Let
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
be two different circles with the radius of same length and centers at points
O
1
O_1
O
1
and
O
2
O_2
O
2
, respectively. Circles
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
are tangent at point
P
P
P
. The line
ℓ
\ell
ℓ
passing through
O
1
O_1
O
1
is tangent to
Γ
2
\Gamma_2
Γ
2
at point
A
A
A
. The line
ℓ
\ell
ℓ
intersects
Γ
1
\Gamma_1
Γ
1
at point
X
X
X
with
X
X
X
between
A
A
A
and
O
1
O_1
O
1
. Let
M
M
M
be the midpoint of
A
X
AX
A
X
and
Y
Y
Y
the intersection of
P
M
PM
PM
and
Γ
2
\Gamma_2
Γ
2
with
Y
≠
P
Y\ne P
Y
=
P
. Prove that
X
Y
XY
X
Y
is parallel to
O
1
O
2
O_1O_2
O
1
O
2
.
2019.5
1
Hide problems
concyclic wanted, circumcircle, angle bisectors midpoint related
Given triangle
A
B
C
ABC
A
BC
, with
A
C
>
B
C
AC> BC
A
C
>
BC
, and the it's circumcircle centered at
O
O
O
. Let
M
M
M
be the point on the circumcircle of triangle
A
B
C
ABC
A
BC
so that
C
M
CM
CM
is the bisector of
∠
A
C
B
\angle ACB
∠
A
CB
. Let
Γ
\Gamma
Γ
be a circle with diameter
C
M
CM
CM
. The bisector of
B
O
C
BOC
BOC
and bisector of
A
O
C
AOC
A
OC
intersect
Γ
\Gamma
Γ
at
P
P
P
and
Q
Q
Q
, respectively. If
K
K
K
is the midpoint of
C
M
CM
CM
, prove that
P
,
Q
,
O
,
K
P, Q, O, K
P
,
Q
,
O
,
K
lie at one point of the circle.
2019.1
1
Hide problems
Easy 3D geometry problem
Given cube
A
B
C
D
.
E
F
G
H
ABCD.EFGH
A
BC
D
.
EFG
H
with
A
B
=
4
AB = 4
A
B
=
4
and
P
P
P
midpoint of the side
E
F
G
H
EFGH
EFG
H
. If
M
M
M
is the midpoint of
P
H
PH
P
H
, find the length of segment
A
M
AM
A
M
.
2016.4
1
Hide problems
Geometry proving parallel
Let
P
A
PA
P
A
and
P
B
PB
PB
be the tangent of a circle
ω
\omega
ω
from a point
P
P
P
outside the circle. Let
M
M
M
be any point on
A
P
AP
A
P
and
N
N
N
is the midpoint of segment
A
B
AB
A
B
.
M
N
MN
MN
cuts
ω
\omega
ω
at
C
C
C
such that
N
N
N
is between
M
M
M
and
C
C
C
. Suppose
P
C
PC
PC
cuts
ω
\omega
ω
at
D
D
D
and
N
D
ND
N
D
cuts
P
B
PB
PB
at
Q
Q
Q
. Prove
M
Q
MQ
MQ
is parallel to
A
B
AB
A
B
.
2008.3
1
Hide problems
Triangle
Given triangle
A
B
C
ABC
A
BC
. The incircle of triangle
A
B
C
ABC
A
BC
is tangent to
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
at
D
,
E
,
F
D,E,F
D
,
E
,
F
respectively. Construct point
G
G
G
on
E
F
EF
EF
such that
D
G
DG
D
G
is perpendicular to
E
F
EF
EF
. Prove that \frac{FG}{EG} \equal{} \frac{BF}{CE}.