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National and Regional Contests
Switzerland Contests
Swiss NMO - geometry
Swiss NMO - geometry
Part of
Switzerland Contests
Subcontests
(47)
2022.8
1
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AX is independent of the choice of P, XI_1 _|_ XI_2, 2 incenters
Let
A
B
C
ABC
A
BC
be a triangle and let
P
P
P
be a point in the interior of the side
B
C
BC
BC
. Let
I
1
I_1
I
1
and
I
2
I_2
I
2
be the incenters of the triangles
A
P
B
AP B
A
PB
and
A
P
C
AP C
A
PC
, respectively. Let
X
X
X
be the closest point to
A
A
A
on the line
A
P
AP
A
P
such that
X
I
1
XI_1
X
I
1
is perpendicular to
X
I
2
XI_2
X
I
2
. Prove that the distance
A
X
AX
A
X
is independent of the choice of
P
P
P
.
2022.1
1
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BX/XD=? AB diameter of circle , AC = AM, CD = AB
Let
k
k
k
be a circle with centre
M
M
M
and let
A
B
AB
A
B
be a diameter of
k
k
k
. Furthermore, let
C
C
C
be a point on
k
k
k
such that
A
C
=
A
M
AC = AM
A
C
=
A
M
. Let
D
D
D
be the point on the line
A
C
AC
A
C
such that
C
D
=
A
B
CD = AB
C
D
=
A
B
and
C
C
C
lies between
A
A
A
and
D
D
D
. Let
E
E
E
be the second intersection of the circumcircle of
B
C
D
BCD
BC
D
with line
A
B
AB
A
B
and
F
F
F
be the intersection of the lines
E
D
ED
E
D
and
B
C
BC
BC
. The line
A
F
AF
A
F
cuts the segment
B
D
BD
B
D
in
X
X
X
. Determine the ratio
B
X
/
X
D
BX/XD
BX
/
X
D
.
2021.8
1
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Geometry from YouTube
Let
△
A
B
C
\triangle ABC
△
A
BC
be a triangle with
A
B
=
A
C
AB =AC
A
B
=
A
C
and
∠
B
A
C
=
2
0
∘
\angle BAC = 20^{\circ}
∠
B
A
C
=
2
0
∘
. Let
D
D
D
be point on the side
A
B
AB
A
B
such that
∠
B
C
D
=
7
0
∘
\angle BCD = 70^{\circ}
∠
BC
D
=
7
0
∘
. Let
E
E
E
be point on the side
A
C
AC
A
C
such that
∠
C
B
E
=
6
0
∘
\angle CBE = 60^{\circ}
∠
CBE
=
6
0
∘
. Determine the value of angle
∠
C
D
E
\angle CDE
∠
C
D
E
.
2021.2
1
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Angle chasing problem
Let
△
A
B
C
\triangle ABC
△
A
BC
be an acute triangle with
A
B
=
A
C
AB =AC
A
B
=
A
C
and let
D
D
D
be a point on the side
B
C
BC
BC
. The circle with centre
D
D
D
passing through
C
C
C
intersects
⊙
(
A
B
D
)
\odot(ABD)
⊙
(
A
B
D
)
at points
P
P
P
and
Q
Q
Q
, where
Q
Q
Q
is the point closer to
B
B
B
. The line
B
Q
BQ
BQ
intersects
A
D
AD
A
D
and
A
C
AC
A
C
at points
X
X
X
and
Y
Y
Y
respectively. Prove that quadrilateral
P
D
X
Y
PDXY
P
D
X
Y
is cyclic.
2020.7
1
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cyclic wanted, isosceles trapezoid, // to angle bisectors related
Let
A
B
C
D
ABCD
A
BC
D
be an isosceles trapezoid with bases
A
D
>
B
C
AD> BC
A
D
>
BC
. Let
X
X
X
be the intersection of the bisectors of
∠
B
A
C
\angle BAC
∠
B
A
C
and
B
C
BC
BC
. Let
E
E
E
be the intersection of
D
B
DB
D
B
with the parallel to the bisector of
∠
C
B
D
\angle CBD
∠
CB
D
through
X
X
X
and let
F
F
F
be the intersection of
D
C
DC
D
C
with the parallel to the bisector of
∠
D
C
B
\angle DCB
∠
D
CB
through
X
X
X
. Show that quadrilateral
A
E
F
D
AEFD
A
EF
D
is cyclic.
2020.2
1
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concurrent wanted, midpoints , arc midpoints of circumcircle related
Let
A
B
C
ABC
A
BC
be an acute triangle. Let
M
A
,
M
B
M_A, M_B
M
A
,
M
B
and
M
C
M_C
M
C
be the midpoints of sides
B
C
,
C
A
BC,CA
BC
,
C
A
, respectively
A
B
AB
A
B
. Let
M
A
′
,
M
B
′
M'_A , M'_B
M
A
′
,
M
B
′
and
M
C
′
M'_C
M
C
′
be the the midpoints of the arcs
B
C
,
C
A
BC, CA
BC
,
C
A
and
A
B
AB
A
B
respectively of the circumscriberd circle of triangle
A
B
C
ABC
A
BC
. Let
P
A
P_A
P
A
be the intersection of the straight line
M
B
M
C
M_BM_C
M
B
M
C
and the perpendicular to
M
B
′
M
C
′
M'_BM'_C
M
B
′
M
C
′
through
A
A
A
. Define
P
B
P_B
P
B
and
P
C
P_C
P
C
similarly. Show that the straight line
M
A
P
A
,
M
B
P
B
M_AP_A, M_BP_B
M
A
P
A
,
M
B
P
B
and
M
C
P
C
M_CP_C
M
C
P
C
intersect at one point.
2004.9
1
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concurrent wanted, ABCD cyclic with AB + CD = BC
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral, so that
∣
A
B
∣
+
∣
C
D
∣
=
∣
B
C
∣
|AB| + |CD| = |BC|
∣
A
B
∣
+
∣
C
D
∣
=
∣
BC
∣
. Show that the intersection of the bisector of
∠
D
A
B
\angle DAB
∠
D
A
B
and
∠
C
D
A
\angle CDA
∠
C
D
A
lies on the side
B
C
BC
BC
.
2004.1
1
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isosceles wanted, tangent and secant from exterior point of a circle
Let
Γ
\Gamma
Γ
be a circle and
P
P
P
a point outside of
Γ
\Gamma
Γ
. A tangent from
P
P
P
to the circle intersects it in
A
A
A
. Another line through
P
P
P
intersects
Γ
\Gamma
Γ
at the points
B
B
B
and
C
C
C
. The bisector of
∠
A
P
B
\angle APB
∠
A
PB
intersects
A
B
AB
A
B
at
D
D
D
and
A
C
AC
A
C
at
E
E
E
. Prove that the triangle
A
D
E
ADE
A
D
E
is isosceles.
2005.8
1
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orthocenter collinear with intersections of circles of diameter BN,CM
Let
A
B
C
ABC
A
BC
be an acute-angled triangle.
M
,
N
M ,N
M
,
N
are any two points on the sides
A
B
,
A
C
AB , AC
A
B
,
A
C
respectively. The circles with the diameters
B
N
BN
BN
and
C
M
CM
CM
intersect at points
P
P
P
and
Q
Q
Q
. Show that the points
P
,
Q
P, Q
P
,
Q
and the orthocenter of the triangle
A
B
C
ABC
A
BC
lie on a straight line.
2005.1
1
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equilateral criterion with medians and 2 cyclic quadrilaterals
Let
A
B
C
ABC
A
BC
be any triangle and
D
,
E
,
F
D, E, F
D
,
E
,
F
the midpoints of
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
. The medians
A
D
,
B
E
AD, BE
A
D
,
BE
and
C
F
CF
CF
intersect at point
S
S
S
. At least two of the quadrilaterals
A
F
S
E
,
B
D
S
F
,
C
E
S
D
AF SE, BDSF, CESD
A
FSE
,
B
D
SF
,
CES
D
are cyclic. Show that the triangle
A
B
C
ABC
A
BC
is equilateral.
2006.7
1
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CD + DA = AB if ABCD cyclic, <ABC=60^o, BC=CD
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral with
∠
A
B
C
=
6
0
o
\angle ABC = 60^o
∠
A
BC
=
6
0
o
and
∣
B
C
∣
=
∣
C
D
∣
| BC | = | CD |
∣
BC
∣
=
∣
C
D
∣
. Prove that
∣
C
D
∣
+
∣
D
A
∣
=
∣
A
B
∣
|CD| + |DA| = |AB|
∣
C
D
∣
+
∣
D
A
∣
=
∣
A
B
∣
2006.5
1
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concyclic wanted, 2 circles internally tangent
A circle
k
1
k_1
k
1
lies within a second circle
k
2
k_2
k
2
and touches it at point
A
A
A
. A line through
A
A
A
intersects
k
1
k_1
k
1
again in
B
B
B
and
k
2
k_2
k
2
in
C
C
C
. The tangent to
k
1
k_1
k
1
through
B
B
B
intersects
k
2
k_2
k
2
at points
D
D
D
and
E
E
E
. The tangents at
k
1
k_1
k
1
passing through
C
C
C
intersects
k
1
k_1
k
1
in points
F
F
F
and
G
G
G
. Prove that
D
,
E
,
F
D, E, F
D
,
E
,
F
and
G
G
G
lie on a circle.
2006.2
1
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| BD | + | AM | + | AN | = | CD | + | AP | + | AQ |, equilateral and circle
Let
A
B
C
ABC
A
BC
be an equilateral triangle and let
D
D
D
be an inner point of the side
B
C
BC
BC
. A circle is tangent to
B
C
BC
BC
at
D
D
D
and intersects the sides
A
B
AB
A
B
and
A
C
AC
A
C
in the inner points
M
,
N
M, N
M
,
N
and
P
,
Q
P, Q
P
,
Q
respectively. Prove that
∣
B
D
∣
+
∣
A
M
∣
+
∣
A
N
∣
=
∣
C
D
∣
+
∣
A
P
∣
+
∣
A
Q
∣
|BD| + |AM| + |AN| = |CD| + |AP| + |AQ|
∣
B
D
∣
+
∣
A
M
∣
+
∣
A
N
∣
=
∣
C
D
∣
+
∣
A
P
∣
+
∣
A
Q
∣
.
2007.6
1
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paralellogram wanted, 3 equal circles intersecing in pairs
Three equal circles
k
1
,
k
2
,
k
3
k_1, k_2, k_3
k
1
,
k
2
,
k
3
intersect non-tangentially at a point
P
P
P
. Let
A
A
A
and
B
B
B
be the centers of circles
k
1
k_1
k
1
and
k
2
k_2
k
2
. Let
D
D
D
and
C
C
C
be the intersection of
k
3
k_3
k
3
with
k
1
k_1
k
1
and
k
2
k_2
k
2
respectively, which is different from
P
P
P
. Show that
A
B
C
D
ABCD
A
BC
D
is a parallelogram.
2007.4
1
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perpendicular wanted, orthocenter related
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with
A
B
>
A
C
AB> AC
A
B
>
A
C
and orthocenter
H
H
H
. Let
D
D
D
the projection of
A
A
A
on
B
C
BC
BC
. Let
E
E
E
be the reflection of
C
C
C
wrt
D
D
D
. The lines
A
E
AE
A
E
and
B
H
BH
B
H
intersect at point
S
S
S
. Let
N
N
N
be the midpoint of
A
E
AE
A
E
and let
M
M
M
be the midpoint of
B
H
BH
B
H
. Prove that
M
N
MN
MN
is perpendicular to
D
S
DS
D
S
.
2008.8
1
Hide problems
diagonals' concurrency criterion in a cyclic hexagon
Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a convex hexagon inscribed in a circle . Prove that the diagonals
A
D
,
B
E
AD, BE
A
D
,
BE
and
C
F
CF
CF
intersect at one point if and only if
A
B
B
C
⋅
C
D
D
E
⋅
E
F
F
A
=
1
\frac{AB}{BC} \cdot \frac{CD}{DE}\cdot \frac{EF}{FA}=1
BC
A
B
⋅
D
E
C
D
⋅
F
A
EF
=
1
2008.5
1
Hide problems
locus of AP x CP + BPx DP = 1 where ABCD square of sidelength 1
Let
A
B
C
D
ABCD
A
BC
D
be a square with side length
1
1
1
. Find the locus of all points
P
P
P
with the property
A
P
⋅
C
P
+
B
P
⋅
D
P
=
1
AP\cdot CP + BP\cdot DP = 1
A
P
⋅
CP
+
BP
⋅
D
P
=
1
.
2008.1
1
Hide problems
a square has equal area with a triangle, 2 circumcenters related
Let
A
B
C
ABC
A
BC
be a triangle with
∠
B
A
C
≠
4
5
o
\angle BAC \ne 45^o
∠
B
A
C
=
4
5
o
and
∠
A
B
C
≠
13
5
o
\angle ABC \ne 135^o
∠
A
BC
=
13
5
o
. Let
P
P
P
be the point on the line
A
B
AB
A
B
with
∠
C
P
B
=
4
5
o
\angle CPB = 45^o
∠
CPB
=
4
5
o
. Let
O
1
O_1
O
1
and
O
2
O_2
O
2
be the centers of the circumcircles of the triangles
A
C
P
ACP
A
CP
and
B
C
P
BCP
BCP
respectively. Show that the area of the square
C
O
1
P
O
2
CO_1P O_2
C
O
1
P
O
2
is equal to the area of the triangle
A
B
C
ABC
A
BC
.
2009.7
1
Hide problems
concurrent wanted, common tangent of intersecting circles related
Points
A
,
M
1
,
M
2
A, M_1, M_2
A
,
M
1
,
M
2
and
C
C
C
are on a line in this order. Let
k
1
k_1
k
1
the circle with center
M
1
M_1
M
1
passing through
A
A
A
and
k
2
k_2
k
2
the circle with center
M
2
M_2
M
2
passing through
C
C
C
. The two circles intersect at points
E
E
E
and
F
F
F
. A common tangent of
k
1
k_1
k
1
and
k
2
k_2
k
2
, touches
k
1
k_1
k
1
at
B
B
B
and
k
2
k_2
k
2
at
D
D
D
. Show that the lines
A
B
,
C
D
AB, CD
A
B
,
C
D
and
E
F
EF
EF
intersect at one point.
2009.5
1
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IM bisects segment AD, incircle related
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
≠
A
C
AB \ne AC
A
B
=
A
C
and incenter
I
I
I
. The incircle touches
B
C
BC
BC
at
D
D
D
. Let
M
M
M
be the midpoint of
B
C
BC
BC
. Show that the line
I
M
IM
I
M
bisects segment
A
D
AD
A
D
.
2012.10
1
Hide problems
concyclic wanted, projections of interior point on sides of ABC related
Let
O
O
O
be an inner point of an acute-angled triangle
A
B
C
ABC
A
BC
. Let
A
1
,
B
1
A_1, B_1
A
1
,
B
1
and
C
1
C_1
C
1
be the projections of
O
O
O
on the sides
B
C
,
A
C
BC, AC
BC
,
A
C
and
A
B
AB
A
B
respectively . Let
P
P
P
be the intersection of the perpendiculars on
B
1
C
1
B_1C_1
B
1
C
1
and
A
1
C
1
A_1C_1
A
1
C
1
from points
A
A
A
and
B
B
B
respectilvey. Let
H
H
H
be the projection of
P
P
P
on
A
B
AB
A
B
. Show that points
A
1
,
B
1
,
C
1
A_1, B_1, C_1
A
1
,
B
1
,
C
1
and
H
H
H
lie on a circle.
2012.6
1
Hide problems
equal circumcircles wanted, starting with a parallelogram
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram with at least an angle not equal to
9
0
o
90^o
9
0
o
and
k
k
k
the circumcircle of the triangle
A
B
C
ABC
A
BC
. Let
E
E
E
be the diametrically opposite point of
B
B
B
. Show that the circumcircle of the triangle
A
D
E
ADE
A
D
E
and
k
k
k
have the same radius.
2012.3
1
Hide problems
<DPM = <BDC wanted, common tangent to intersecting circles related
The circles
k
1
k_1
k
1
and
k
2
k_2
k
2
intersect at points
D
D
D
and
P
P
P
. The common tangent of the two circles on the side of
D
D
D
touches
k
1
k_1
k
1
at
A
A
A
and
k
2
k_2
k
2
at
B
B
B
. The straight line
A
D
AD
A
D
intersects
k
2
k_2
k
2
for a second time at
C
C
C
. Let
M
M
M
be the center of the segment
B
C
BC
BC
. Show that
∠
D
P
M
=
∠
B
D
C
\angle DPM = \angle BDC
∠
D
PM
=
∠
B
D
C
.
2013.10
1
Hide problems
concurrency inside a tangential wanted 2 perpendiculars, 1 angle bisector
Let
A
B
C
D
ABCD
A
BC
D
be a tangential quadrilateral with
B
C
>
B
A
BC> BA
BC
>
B
A
. The point
P
P
P
is on the segment
B
C
BC
BC
, such that
B
P
=
B
A
BP = BA
BP
=
B
A
. Show that the bisector of
∠
B
C
D
\angle BCD
∠
BC
D
, the perpendicular on line
B
C
BC
BC
through
P
P
P
and the perpendicular on
B
D
BD
B
D
through
A
A
A
, intersect at one point.
2013.7
1
Hide problems
ST bisects BC, <ASO = <ACO, <ATO =< ABO
Let
O
O
O
be the center of the circle of the triangle
A
B
C
ABC
A
BC
with
A
B
≠
A
C
AB \ne AC
A
B
=
A
C
. Furthermore, let
S
S
S
and
T
T
T
be points on the rays
A
B
AB
A
B
and
A
C
AC
A
C
, such that
∠
A
S
O
=
∠
A
C
O
\angle ASO = \angle ACO
∠
A
SO
=
∠
A
CO
and
∠
A
T
O
=
∠
A
B
O
\angle ATO = \angle ABO
∠
A
TO
=
∠
A
BO
. Show that
S
T
ST
ST
bisects the segment
B
C
BC
BC
.
2013.3
1
Hide problems
BC = DE - BE if <ADC =< DBA, A projection of A on BD , cyclic ABCD
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral with
∠
A
D
C
=
∠
D
B
A
\angle ADC = \angle DBA
∠
A
D
C
=
∠
D
B
A
. Furthermore, let
E
E
E
be the projection of
A
A
A
on
B
D
BD
B
D
. Show that
B
C
=
D
E
−
B
E
BC = DE - BE
BC
=
D
E
−
BE
.
2014.10
1
Hide problems
DB bisects segment CT, tagnet to circle related
Let
k
k
k
be a circle with diameter
A
B
AB
A
B
. Let
C
C
C
be a point on the straight line
A
B
AB
A
B
, so that
B
B
B
between
A
A
A
and
C
C
C
lies. Let
T
T
T
be a point on
k
k
k
such that
C
T
CT
CT
is a tangent to
k
k
k
. Let
l
l
l
be the parallel to
C
T
CT
CT
through
A
A
A
and
D
D
D
the intersection of
l
l
l
and the perpendicular to
A
B
AB
A
B
through
T
T
T
. Show that the line
D
B
DB
D
B
bisects segment
C
T
CT
CT
.
2014.8
1
Hide problems
concyclic wanted, midpoints of altitudes in acute related
In the acute-angled triangle
A
B
C
ABC
A
BC
, let
M
M
M
be the midpoint of the atlitude
h
b
h_b
h
b
through
B
B
B
and
N
N
N
be the midpoint of the height
h
c
h_c
h
c
through
C
C
C
. Further let
P
P
P
be the intersection of
A
M
AM
A
M
and
h
c
h_c
h
c
and
Q
Q
Q
be the intersection of
A
N
AN
A
N
and
h
b
h_b
h
b
. Show that
M
,
N
,
P
M, N, P
M
,
N
,
P
and
Q
Q
Q
lie on a circle.
2014.1
1
Hide problems
GH // t wanted, 4 concyclic points, tangent, reflection of AB wrt AC
The points
A
,
B
,
C
A, B, C
A
,
B
,
C
and
D
D
D
lie in this order on the circle
k
k
k
. Let
t
t
t
be the tangent at
k
k
k
through
C
C
C
and
s
s
s
the reflection of
A
B
AB
A
B
at
A
C
AC
A
C
. Let
G
G
G
be the intersection of the straight line
A
C
AC
A
C
and
B
D
BD
B
D
and
H
H
H
the intersection of the straight lines
s
s
s
and
C
D
CD
C
D
. Show that
G
H
GH
G
H
is parallel to
t
t
t
.
2015.8
1
Hide problems
parallel concurrency , trapezoid related
Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid, where
A
B
AB
A
B
and
C
D
CD
C
D
are parallel. Let
P
P
P
be a point on the side
B
C
BC
BC
. Show that the parallels to
A
P
AP
A
P
and
P
D
PD
P
D
intersect through
C
C
C
and
B
B
B
to
D
A
DA
D
A
, respectively.
2015.4
2
Hide problems
an apollonius circle construction, PPC in swiss NMO
Given a circle
k
k
k
and two points
A
A
A
and
B
B
B
outside the circle. Specify how to can construct a circle with a compass and ruler, so that
A
A
A
and
B
B
B
lie on that circle and that circle is tangent to
k
k
k
.
an apollonius circle construction, PPC in swiss NMO
Given a circle
k
k
k
and two points
A
A
A
and
B
B
B
outside the circle. Specify how to can construct a circle with a compass and ruler, so that
A
A
A
and
B
B
B
lie on that circle and that circle is tangent to
k
k
k
.
2015.1
1
Hide problems
AD: DC, interesections of internal and external bisectors of C with circumcircle
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with
A
B
≠
B
C
AB \ne BC
A
B
=
BC
and radius
k
k
k
. Let
P
P
P
and
Q
Q
Q
be the points of intersection of
k
k
k
with the internal bisector and the external bisector of
∠
C
B
A
\angle CBA
∠
CB
A
respectively. Let
D
D
D
be the intersection of
A
C
AC
A
C
and
P
Q
PQ
PQ
. Find the ratio
A
D
:
D
C
AD: DC
A
D
:
D
C
.
2016.8
1
Hide problems
IJ = AH wanted, parallel , starting with northocenter of an acute triangle
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with height intersection
H
H
H
. Let
G
G
G
be the intersection of parallel of
A
B
AB
A
B
through
H
H
H
with the parallel of
A
H
AH
A
H
through
B
B
B
. Let
I
I
I
be the point on the line
G
H
GH
G
H
, so that
A
C
AC
A
C
bisects segment
H
I
HI
H
I
. Let
J
J
J
be the second intersection of
A
C
AC
A
C
and the circumcircle of the triangle
C
G
I
CGI
CG
I
. Show that
I
J
=
A
H
IJ = AH
I
J
=
A
H
2016.5
1
Hide problems
2 circles and 1 line concurrent, <ABC=90^o, <CPA = < BAC, <BQC = < CBA
Let
A
B
C
ABC
A
BC
be a right triangle with
∠
A
C
B
=
9
0
o
\angle ACB = 90^o
∠
A
CB
=
9
0
o
and M the center of
A
B
AB
A
B
. Let
G
G
G
br any point on the line
M
C
MC
MC
and
P
P
P
a point on the line
A
G
AG
A
G
, such that
∠
C
P
A
=
∠
B
A
C
\angle CPA = \angle BAC
∠
CP
A
=
∠
B
A
C
. Further let
Q
Q
Q
be a point on the straight line
B
G
BG
BG
, such that
∠
B
Q
C
=
∠
C
B
A
\angle BQC = \angle CBA
∠
BQC
=
∠
CB
A
. Show that the circles of the triangles
A
Q
G
AQG
A
QG
and
B
P
G
BPG
BPG
intersect on the segment
A
B
AB
A
B
.
2016.1
1
Hide problems
angle chasing , <BAC = 60^o , 2 < BAE = <ACB
Let
A
B
C
ABC
A
BC
be a triangle with
∠
B
A
C
=
6
0
o
\angle BAC = 60^o
∠
B
A
C
=
6
0
o
. Let
E
E
E
be the point on the side
B
C
BC
BC
, such that
2
∠
B
A
E
=
∠
A
C
B
2 \angle BAE = \angle ACB
2∠
B
A
E
=
∠
A
CB
. Let
D
D
D
be the second intersection of
A
B
AB
A
B
and the circumcircle of the triangle
A
E
C
AEC
A
EC
and
P
P
P
be the second intersection of
C
D
CD
C
D
and the circumcircle of the triangle
D
B
E
DBE
D
BE
. Calculate the angle
∠
B
A
P
\angle BAP
∠
B
A
P
.
2017.8
1
Hide problems
midpoint wanted, staring with an isosceles ABC and circle (A, AB=AC)
Let
A
B
C
ABC
A
BC
be an isosceles triangle with vertex
A
A
A
and
A
B
>
B
C
AB> BC
A
B
>
BC
. Let
k
k
k
be the circle with center
A
A
A
passsing through
B
B
B
and
C
C
C
. Let
H
H
H
be the second intersection of
k
k
k
with the altitude of the triangle
A
B
C
ABC
A
BC
through
B
B
B
. Further let
G
G
G
be the second intersection of
k
k
k
with the median through
B
B
B
in triangle
A
B
C
ABC
A
BC
. Let
X
X
X
be the intersection of the lines
A
C
AC
A
C
and
G
H
GH
G
H
. Show that
C
C
C
is the midpoint of
A
X
AX
A
X
.
2017.5
1
Hide problems
BR = 2XY wanted, AR = BP, AQ = AB ,tangent to circumcircle related
Let
A
B
C
ABC
A
BC
be a triangle with
A
C
>
A
B
AC> AB
A
C
>
A
B
. Let
P
P
P
be the intersection of
B
C
BC
BC
and the tangent through
A
A
A
around the triangle
A
B
C
ABC
A
BC
. Let
Q
Q
Q
be the point on the straight line
A
C
AC
A
C
, so that
A
Q
=
A
B
AQ = AB
A
Q
=
A
B
and
A
A
A
is between
C
C
C
and
Q
Q
Q
. Let
X
X
X
and
Y
Y
Y
be the center of
B
Q
BQ
BQ
and
A
P
AP
A
P
. Let
R
R
R
be the point on
A
P
AP
A
P
so that
A
R
=
B
P
AR = BP
A
R
=
BP
and
R
R
R
is between
A
A
A
and
P
P
P
. Show that
B
R
=
2
X
Y
BR = 2XY
BR
=
2
X
Y
.
2017.1
1
Hide problems
AD = AO wanted, 2 circles and angle bisector related
Let
A
A
A
and
B
B
B
be points on the circle
k
k
k
with center
O
O
O
, so that
A
B
>
A
O
AB> AO
A
B
>
A
O
. Let
C
C
C
be the intersection of the bisectors of
∠
O
A
B
\angle OAB
∠
O
A
B
and
k
k
k
, different from
A
A
A
. Let
D
D
D
be the intersection of the straight line
A
B
AB
A
B
with the circumcircle of the triangle
O
B
C
OBC
OBC
, different from
B
B
B
. Show that
A
D
=
A
O
AD = AO
A
D
=
A
O
.
2018.6
1
Hide problems
BD = EF wanted, BE// FG incircle related
Let
k
k
k
be the incircle of the triangle
A
B
C
ABC
A
BC
with the center of the incircle
I
I
I
. The circle
k
k
k
touches the sides
B
C
,
C
A
BC, CA
BC
,
C
A
and
A
B
AB
A
B
in points
D
,
E
D, E
D
,
E
and
F
F
F
. Let
G
G
G
be the intersection of the straight line
A
I
AI
A
I
and the circle
k
k
k
, which lies between
A
A
A
and
I
I
I
. Assume
B
E
BE
BE
and
F
G
FG
FG
are parallel. Show that
B
D
=
E
F
BD = EF
B
D
=
EF
.
2018.4
1
Hide problems
concurrency wanted, <BAD = <DBC, <DAC = <BCD
Let
D
D
D
be a point inside an acute triangle
A
B
C
ABC
A
BC
, such that
∠
B
A
D
=
∠
D
B
C
\angle BAD = \angle DBC
∠
B
A
D
=
∠
D
BC
and
∠
D
A
C
=
∠
B
C
D
\angle DAC = \angle BCD
∠
D
A
C
=
∠
BC
D
. Let
P
P
P
be a point on the circumcircle of the triangle
A
D
B
ADB
A
D
B
. Suppose
P
P
P
are itself outside the triangle
A
B
C
ABC
A
BC
. A line through
P
P
P
intersects the ray
B
A
BA
B
A
in
X
X
X
and ray
C
A
CA
C
A
in
Y
Y
Y
, so that
∠
X
P
B
=
∠
P
D
B
\angle XPB = \angle PDB
∠
XPB
=
∠
P
D
B
. Show that
B
Y
BY
B
Y
and
C
X
CX
CX
intersect on
A
D
AD
A
D
.
2019.7
1
Hide problems
<ACB = 3 <DCB , if <CAB = 2 <ABC, AD = BD, CD = AC
Let
A
B
C
ABC
A
BC
be a triangle with
∠
C
A
B
=
2
∠
A
B
C
\angle CAB = 2 \angle ABC
∠
C
A
B
=
2∠
A
BC
. Assume that a point
D
D
D
is inside the triangle
A
B
C
ABC
A
BC
exists such that
A
D
=
B
D
AD = BD
A
D
=
B
D
and
C
D
=
A
C
CD = AC
C
D
=
A
C
. Show that
∠
A
C
B
=
3
∠
D
C
B
\angle ACB = 3 \angle DCB
∠
A
CB
=
3∠
D
CB
.
2019.1
1
Hide problems
fixed point, intersection of line with circle
Let
A
A
A
be a point and let k be a circle through
A
A
A
. Let
B
B
B
and
C
C
C
be two more points on
k
k
k
. Let
X
X
X
be the intersection of the bisector of
∠
A
B
C
\angle ABC
∠
A
BC
with
k
k
k
. Let
Y
Y
Y
be the reflection of
A
A
A
wrt point
X
X
X
, and
D
D
D
the intersection of the straight line
Y
C
YC
Y
C
with
k
k
k
. Prove that point
D
D
D
is independent of the choice of
B
B
B
and
C
C
C
on the circle
k
k
k
.
2011.8
1
Hide problems
Concyclic Points - Switzerland 2011
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram and
H
H
H
the Orthocentre of
△
A
B
C
\triangle{ABC}
△
A
BC
. The line parallel to
A
B
AB
A
B
through
H
H
H
intersects
B
C
BC
BC
at
P
P
P
and
A
D
AD
A
D
at
Q
Q
Q
while the line parallel to
B
C
BC
BC
through
H
H
H
intersects
A
B
AB
A
B
at
R
R
R
and
C
D
CD
C
D
at
S
S
S
. Show that
P
P
P
,
Q
Q
Q
,
R
R
R
and
S
S
S
are concyclic.(Swiss Mathematical Olympiad 2011, Final round, problem 8)
2011.5
1
Hide problems
Ratio - Switzerland 2011
Let
△
A
B
C
\triangle{ABC}
△
A
BC
be a triangle with circumcircle
τ
\tau
τ
. The tangentlines to
τ
\tau
τ
through
A
A
A
and
B
B
B
intersect at
T
T
T
. The circle through
A
A
A
,
B
B
B
and
T
T
T
intersects
B
C
BC
BC
and
A
C
AC
A
C
again at
D
D
D
and
E
E
E
, respectively;
C
T
CT
CT
and
B
E
BE
BE
intersect at
F
F
F
. Suppose
D
D
D
is the midpoint of
B
C
BC
BC
. Calculate the ratio
B
F
:
B
E
BF:BE
BF
:
BE
.(Swiss Mathematical Olympiad 2011, Final round, problem 5)
2011.2
1
Hide problems
Feet of perpendiculars - Switzerland 2011
Let
△
A
B
C
\triangle{ABC}
△
A
BC
be an acute-angled triangle and let
D
D
D
,
E
E
E
,
F
F
F
be points on
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
, respectively, such that \angle{AFE}=\angle{BFD}\mbox{,} \angle{BDF}=\angle{CDE} \mbox{and} \angle{CED}=\angle{AEF}\mbox{.} Prove that
D
D
D
,
E
E
E
and
F
F
F
are the feet of the perpendiculars through
A
A
A
,
B
B
B
and
C
C
C
on
B
C
BC
BC
,
C
A
CA
C
A
and
A
B
AB
A
B
, respectively.(Swiss Mathematical Olympiad 2011, Final round, problem 2)
2010.9
1
Hide problems
Three circles have a common point
Let
k
k
k
and
k
′
k'
k
′
two concentric circles centered at
O
O
O
, with
k
′
k'
k
′
being larger than
k
k
k
. A line through
O
O
O
intersects
k
k
k
at
A
A
A
and
k
′
k'
k
′
at
B
B
B
such that
O
O
O
seperates
A
A
A
and
B
B
B
. Another line through
O
O
O
intersects
k
k
k
at
E
E
E
and
k
′
k'
k
′
at
F
F
F
such that
E
E
E
separates
O
O
O
and
F
F
F
. Show that the circumcircle of
△
O
A
E
\triangle{OAE}
△
O
A
E
and the circles with diametres
A
B
AB
A
B
and
E
F
EF
EF
have a common point.
2010.2
1
Hide problems
4 concylic points
Let
△
A
B
C
\triangle{ABC}
△
A
BC
be a triangle with AB\not\equal{}AC. The incircle with centre
I
I
I
touches
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
at
D
D
D
,
E
E
E
,
F
F
F
, respectively. Furthermore let
M
M
M
the midpoint of
E
F
EF
EF
and
A
D
AD
A
D
intersect the incircle at P\not\equal{}D. Show that
P
M
I
D
PMID
PM
I
D
ist cyclic.