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Kyiv City MO
Kyiv City MO - geometry
Kyiv City MO Seniors Round2 2010+ geometry
Kyiv City MO Seniors Round2 2010+ geometry
Part of
Kyiv City MO - geometry
Subcontests
(23)
2021.10.4.1
1
Hide problems
OK// AB wanted in isosceles trapezoid (2021 Kyiv City MO Round2 10.4.1)
Let
A
B
C
D
ABCD
A
BC
D
be an isosceles trapezoid,
A
D
=
B
C
AD=BC
A
D
=
BC
,
A
B
∥
C
D
AB \parallel CD
A
B
∥
C
D
. The diagonals of the trapezoid intersect at the point
O
O
O
, and the point
M
M
M
is the midpoint of the side
A
D
AD
A
D
. The circle circumscribed around the triangle
B
C
M
BCM
BCM
intersects the side
A
D
AD
A
D
at the point
K
K
K
. Prove that
O
K
∥
A
B
OK \parallel AB
O
K
∥
A
B
.
2021.11.3
1
Hide problems
circle tangent to incircle, <BKA=45^o (2021 Kyiv City MO Round2 11.3)
In the triangle
A
B
C
ABC
A
BC
, the altitude
B
H
BH
B
H
and the angle bisector
B
L
BL
B
L
are drawn, the inscribed circle
w
w
w
touches the side of the
A
C
AC
A
C
at the point
K
K
K
. It is known that
∠
B
K
A
=
4
5
o
\angle BKA = 45^o
∠
B
K
A
=
4
5
o
. Prove that the circle with diameter
H
L
HL
H
L
touches the circle
w
w
w
.(Anton Trygub)
2021.11.3.1
1
Hide problems
product of radii, common tangents to circles (2021 Kyiv City MO Round2 11.3.1)
Two circles
k
1
k_1
k
1
and
k
2
k_2
k
2
with radii
r
1
r_1
r
1
and
r
2
r_2
r
2
have no common points. The line
A
B
AB
A
B
is a common internal tangent, and the line
C
D
CD
C
D
is a common external tangent to these circles, where
A
,
C
∈
k
1
A, C \in k_1
A
,
C
∈
k
1
and
B
,
D
∈
k
2
B, D \in k_2
B
,
D
∈
k
2
. Knowing that
A
B
=
12
AB=12
A
B
=
12
and
C
D
=
16
CD =16
C
D
=
16
, find the value of the product
r
1
r
2
r_1r_2
r
1
r
2
.
2021.10.4
1
Hide problems
concyclic, <OAD+<OBC= <ODA + <OCB = 90^o (2021 Kyiv City MO Round2 10.4)
Inside the quadrilateral
A
B
C
D
ABCD
A
BC
D
marked a point
O
O
O
such that
∠
O
A
D
+
∠
O
B
C
=
∠
O
D
A
+
∠
O
C
B
=
9
0
o
\angle OAD+ \angle OBC = \angle ODA + \angle OCB = 90^o
∠
O
A
D
+
∠
OBC
=
∠
O
D
A
+
∠
OCB
=
9
0
o
. Prove that the centers of the circumscribed circles around triangles
O
A
D
OAD
O
A
D
and
O
B
C
OBC
OBC
as well as the midpoints of the sides
A
B
AB
A
B
and
C
D
CD
C
D
lie on one circle.(Anton Trygub)
2020.11.2
1
Hide problems
<TSC=<BAC wanted, parallelogram, circumcircle (2020 Kyiv City MO Round2 11.2)
A point
P
P
P
was chosen on the smaller arc
B
C
BC
BC
of the circumcircle of the acute-angled triangle
A
B
C
ABC
A
BC
. Points
R
R
R
and
S
S
S
on the sides
A
B
AB
A
B
and
A
C
AC
A
C
are respectively selected so that
C
P
R
S
CPRS
CPRS
is a parallelogram. Point
T
T
T
on the arc
A
C
AC
A
C
of the circumscribed circle of
△
A
B
C
\vartriangle ABC
△
A
BC
such that
B
T
∥
C
P
BT \parallel CP
BT
∥
CP
. Prove that
∠
T
S
C
=
∠
B
A
C
\angle TSC = \angle BAC
∠
TSC
=
∠
B
A
C
.(Anton Trygub)
2020.10.2
1
Hide problems
<PBM+<CBM=<PCA,<BM=90^o, <ABC+<APC=180^o (2020 Kyiv City MO Round2 10.2)
Let
M
M
M
be the midpoint of the side
A
C
AC
A
C
of triangle
A
B
C
ABC
A
BC
. Inside
△
B
M
C
\vartriangle BMC
△
BMC
was found a point
P
P
P
such that
∠
B
M
P
=
9
0
o
\angle BMP = 90^o
∠
BMP
=
9
0
o
,
∠
A
B
C
+
∠
A
P
C
=
18
0
o
\angle ABC+ \angle APC =180^o
∠
A
BC
+
∠
A
PC
=
18
0
o
. Prove that
∠
P
B
M
+
∠
C
B
M
=
∠
P
C
A
\angle PBM + \angle CBM = \angle PCA
∠
PBM
+
∠
CBM
=
∠
PC
A
.(Anton Trygub)
2019.11.3.1
1
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centroid is incenter of other triangle, (2019 Kyiv City MO Round2 11.3.1)
It is known that in the triangle
A
B
C
ABC
A
BC
the smallest side is
B
C
BC
BC
. Let
X
,
Y
,
K
X, Y, K
X
,
Y
,
K
and
L
L
L
- points on the sides
A
B
,
A
C
AB, AC
A
B
,
A
C
and on the rays
C
B
,
B
C
CB, BC
CB
,
BC
, respectively, are such that
B
X
=
B
K
=
B
C
=
C
Y
=
C
L
BX = BK = BC =CY =CL
BX
=
B
K
=
BC
=
C
Y
=
C
L
. The line
K
X
KX
K
X
intersects the line
L
Y
LY
L
Y
at the point
M
M
M
. Prove that the intersection point of the medians
△
K
L
M
\vartriangle KLM
△
K
L
M
coincides with the center of the inscribed circle
△
A
B
C
\vartriangle ABC
△
A
BC
.
2019.11.3
1
Hide problems
circumcenter lies on midline of other triangle (2019 Kyiv City MO Round2 11.3)
The line
ℓ
\ell
ℓ
is perpendicular to the side
A
C
AC
A
C
of the acute triangle
A
B
C
ABC
A
BC
and intersects this side at point
K
K
K
, and the circumcribed circle
△
A
B
C
\vartriangle ABC
△
A
BC
at points
P
P
P
and
T
T
T
(point P on the other side of line
A
C
AC
A
C
, as the vertex
B
B
B
). Denote by
P
1
P_1
P
1
and
T
1
T_1
T
1
- the projections of the points
P
P
P
and
T
T
T
on line
A
B
AB
A
B
, with the vertices
A
,
B
A, B
A
,
B
belong to the segment
P
1
T
1
P_1T_1
P
1
T
1
. Prove that the center of the circumscribed circle of the
△
P
1
K
T
1
\vartriangle P_1KT_1
△
P
1
K
T
1
lies on a line containing the midline
△
A
B
C
\vartriangle ABC
△
A
BC
, which is parallel to the side
A
C
AC
A
C
. (Anton Trygub)
2019.10.3.1
1
Hide problems
AR = QR wanted inside a regular pentagon (2019 Kyiv City MO Round2 10.3.1)
Let
A
B
C
D
E
ABCDE
A
BC
D
E
be a regular pentagon with center
M
M
M
. Point
P
≠
M
P \ne M
P
=
M
is selected on segment
M
D
MD
M
D
. The circumscribed circle of triangle
A
B
P
ABP
A
BP
intersects the line
A
E
AE
A
E
for second time at point
Q
Q
Q
, and a line that is perpendicular to the
C
D
CD
C
D
and passes through
P
P
P
, for second time at the point
R
R
R
. Prove that
A
R
=
Q
R
AR = QR
A
R
=
QR
.
2019.10.3
1
Hide problems
<T_A T_B T_C= 90^o - 1/2 <ABC if OI//AC (2019 Kyiv City MO Round2 10.3)
Denote in the triangle
A
B
C
ABC
A
BC
by
T
A
,
T
B
,
T
C
T_A,T_B,T_C
T
A
,
T
B
,
T
C
the touch points of the exscribed circles of
△
A
B
C
\vartriangle ABC
△
A
BC
, tangent to sides
B
C
,
A
C
BC, AC
BC
,
A
C
and
A
B
AB
A
B
respectively. Let
O
O
O
be the center of the circumcircle of
△
A
B
C
\vartriangle ABC
△
A
BC
, and
I
I
I
is the center of it's inscribed circle. It is known that
O
I
∥
A
C
OI\parallel AC
O
I
∥
A
C
. Prove that
∠
T
A
T
B
T
C
=
9
0
o
−
1
2
∠
A
B
C
\angle T_A T_B T_C= 90^o - \frac12 \angle ABC
∠
T
A
T
B
T
C
=
9
0
o
−
2
1
∠
A
BC
. (Anton Trygub)
2018.11.2
1
Hide problems
right angle wanted, AB=BC, circumcircle related (2018 Kyiv City MO Round2 11.2)
In the quadrilateral
A
B
C
D
ABCD
A
BC
D
,
A
B
=
B
C
AB = BC
A
B
=
BC
, the point
K
K
K
is the midpoint of the side
C
D
CD
C
D
, the rays
B
K
BK
B
K
and
A
D
AD
A
D
intersect at the point
M
M
M
, the circumscribed circle
Δ
A
B
M
\Delta ABM
Δ
A
BM
intersects the line
A
C
AC
A
C
for the second time at the point
P
P
P
. Prove that
∠
B
K
P
=
90
∘
\angle BKP = 90 {} ^ \circ
∠
B
K
P
=
90
∘
. (Anton Trygub)
2018.10.3.1
1
Hide problems
perpendicular wanted circumcircle related (2018 Kyiv City MO Round2 10.3.1)
The point
O
O
O
is the center of the circumcircle of the acute triangle
A
B
C
ABC
A
BC
. The line
A
C
AC
A
C
intersects the circumscribed circle
Δ
A
B
O
\Delta ABO
Δ
A
BO
for second time at the point
X
X
X
. Prove that
X
O
⊥
B
C
XO \bot BC
XO
⊥
BC
.
2018.10.3
1
Hide problems
circumcircle of HXY equidistant from B,C (2018 Kyiv City MO Round2 10.3)
In the acute triangle
A
B
C
ABC
A
BC
the orthocenter
H
H
H
and the center of the circumscribed circle
O
O
O
were noted. The line
A
O
AO
A
O
intersects the side
B
C
BC
BC
at the point
D
D
D
. A perpendicular drawn to the side
B
C
BC
BC
at the point
D
D
D
intersects the heights from the vertices
B
B
B
and
C
C
C
of the triangle
A
B
C
ABC
A
BC
at the points
X
X
X
and
Y
Y
Y
respectively. Prove that the center of the circumscribed circle
Δ
H
X
Y
\Delta HXY
Δ
H
X
Y
is equidistant from the points
B
B
B
and
C
C
C
.(Danilo Hilko)
2017.11.2
1
Hide problems
ABTN is cyclic iff AB = AK, KT = KC,CN = BK (2017 Kyiv City MO Round2 11.2)
The median
C
M
CM
CM
is drawn in the triangle
A
B
C
ABC
A
BC
intersecting bisector angle
B
L
BL
B
L
at point
O
O
O
. Ray
A
O
AO
A
O
intersects side
B
C
BC
BC
at point
K
K
K
, beyond point
K
K
K
draw the segment
K
T
=
K
C
KT = KC
K
T
=
K
C
. On the ray
B
C
BC
BC
beyond point
C
C
C
draw a segment
C
N
=
B
K
CN = BK
CN
=
B
K
. Prove that is a quadrilateral
A
B
T
N
ABTN
A
BTN
is cyclic if and only if
A
B
=
A
K
AB = AK
A
B
=
A
K
.(Vladislav Yurashev)
2017.10.3
1
Hide problems
EF=diameter of circumcircle of AO_1O_2 (2017 Kyiv City MO Round2 10.3)
Circles
w
1
w_1
w
1
and
w
2
w_2
w
2
with centers at points
O
1
O_1
O
1
and
O
2
O_2
O
2
respectively, intersect at points
A
A
A
and
B
B
B
. A line passing through point
B
B
B
, intersects the circles
w
1
w_1
w
1
and
w
2
w_2
w
2
at points
C
C
C
and
D
D
D
other than
B
B
B
. Tangents to the circles
w
1
w_1
w
1
and
w
2
w_2
w
2
at points
C
C
C
and
D
D
D
intersect at point
E
E
E
. Line
E
A
EA
E
A
intersects the circumscribed circle
w
w
w
of triangle
A
O
1
O
2
AO_1O_2
A
O
1
O
2
at point
F
F
F
. Prove that the length of the segment is
E
F
EF
EF
is equal to the diameter of the circle
w
w
w
.(Vovchenko V., Plotnikov M.)
2016.10.2
1
Hide problems
square construction given 4 collinear (2016 Kyiv City MO Round2 10.2 11.2 )
On the horizontal line from left to right are the points
P
,
Q
,
R
,
S
P, \, \, Q, \, \, R, \, \, S
P
,
Q
,
R
,
S
. Construct a square
A
B
C
D
ABCD
A
BC
D
, for which on the line
A
D
AD
A
D
lies lies the point
P
P
P
, on the line
B
C
BC
BC
lies the point
Q
Q
Q
, on the line
A
B
AB
A
B
lies the point
R
R
R
, on the line
C
D
CD
C
D
lies the point
S
S
S
.
2015.11.2
1
Hide problems
collinear symmetrics wrt midpoints , equilateral (2015 Kyiv City MO Round2 11.2)
The line passing through the center of the equilateral triangle
A
B
C
ABC
A
BC
intersects the lines
A
B
AB
A
B
,
B
C
BC
BC
and
C
A
CA
C
A
at the points
C
1
{{C} _ {1}}
C
1
,
A
1
{{A} _ {1}}
A
1
and
B
1
{{B} _ {1}}
B
1
, respectively. Let
A
2
{{A} _ {2}}
A
2
be a point that is symmetric
A
1
{{A} _ {1}}
A
1
with respect to the midpoint of
B
C
BC
BC
; the points
B
2
{{B} _ {2}}
B
2
and
C
2
{{C} _ {2}}
C
2
are defined similarly. Prove that the points
A
2
{{A} _ {2}}
A
2
,
B
2
{{B} _ {2}}
B
2
and
C
2
{{C} _ {2}}
C
2
lie on the same line tangent to the inscribed circle of the triangle
A
B
C
ABC
A
BC
.(Serdyuk Nazar)
2014.10.4
1
Hide problems
concurrceny wanted, 3 circles related (2014 Kyiv City MO Round2 10.4 11.3)
Three circles are constructed for the triangle
A
B
C
ABC
A
BC
: the circle
w
A
{{w} _ {A}}
w
A
passes through the vertices
B
B
B
and
C
C
C
and intersects the sides
A
B
AB
A
B
and
A
C
AC
A
C
at points
A
1
{{A} _ {1}}
A
1
and
A
2
{{A} _ {2}}
A
2
respectively, the circle
w
B
{{w} _ {B}}
w
B
passes through the vertices
A
A
A
and
C
C
C
and intersects the sides
B
A
BA
B
A
and
B
C
BC
BC
at the points
B
1
{{B} _ {1}}
B
1
and
B
2
{{B} _ {2}}
B
2
,
w
C
{{w} _ {C}}
w
C
passes through the vertices
A
A
A
and
B
B
B
and intersects the sides
C
A
CA
C
A
and
C
B
CB
CB
at the points
C
1
{{C} _ {1}}
C
1
and
C
2
{{C} _ {2}}
C
2
. Let
A
1
A
2
∩
B
1
B
2
=
C
′
{{A} _ {1}} {{A} _ {2}} \cap {{B} _ {1}} {{B} _ {2}} = {C} '
A
1
A
2
∩
B
1
B
2
=
C
′
,
A
1
A
2
∩
C
1
C
2
=
B
′
{{A} _ {1}} {{A} _ {2}} \cap {{C} _ {1}} {{C} _ {2}} = {B} '
A
1
A
2
∩
C
1
C
2
=
B
′
ta
B
1
B
2
∩
C
1
C
2
=
A
′
{ {B} _ {1}} {{B} _ {2}} \cap {{C} _ {1}} {{C} _ {2}} = {A} '
B
1
B
2
∩
C
1
C
2
=
A
′
is Prove that the perpendiculars, which are omitted from the points
A
′
,
B
′
,
C
′
{A} ', \, \, {B}', \, \, {C} '
A
′
,
B
′
,
C
′
to the lines
B
C
BC
BC
,
C
A
CA
C
A
and
A
B
AB
A
B
respectively intersect at one point.(Rudenko Alexander)
2013.11.4
1
Hide problems
concurrency, <APE =<BAC, < CQF =< BCA (2013 Kyiv City MO Round2 11.4)
Let
H
H
H
be the intersection point of the altitudes
A
P
AP
A
P
and
C
Q
CQ
CQ
of the acute-angled triangle
A
B
C
ABC
A
BC
. On its median
B
M
BM
BM
marked points
E
E
E
and
F
F
F
so that
∠
A
P
E
=
∠
B
A
C
\angle APE = \angle BAC
∠
A
PE
=
∠
B
A
C
and
∠
C
Q
F
=
∠
B
C
A
\angle CQF = \angle BCA
∠
CQF
=
∠
BC
A
, and the point
E
E
E
lies inside the triangle
A
P
B
APB
A
PB
, and the point
F
F
F
lies inside the triangle
C
Q
B
CQB
CQB
. Prove that the lines
A
E
AE
A
E
,
C
F
CF
CF
and
B
H
BH
B
H
intersect at one point.(Vyacheslav Yasinsky)
2012.10.4
1
Hide problems
compare angles between altitude-median (2012 Kyiv City MO Round2 10.4)
In the triangle
A
B
C
ABC
A
BC
with sides
B
C
>
A
C
>
A
B
BC> AC> AB
BC
>
A
C
>
A
B
the angles between altiude and median drawn from one vertex are considered. Find out at which vertex this angle is the largest of the three.(Rozhkova Maria)
2012.11.4
1
Hide problems
concurrent or //, // chords of inters. circles (2012 Kyiv City MO Round2 11.4)
The circles
w
1
{{w} _ {1}}
w
1
and
w
2
{{w} _ {2}}
w
2
intersect at points
P
P
P
and
Q
Q
Q
. Let
A
B
AB
A
B
and
C
D
CD
C
D
be parallel diameters of circles
w
1
{ {w} _ {1}}
w
1
and
w
2
{{w} _ {2}}
w
2
, respectively. In this case, none of the points
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
coincides with either
P
P
P
or
Q
Q
Q
, and the points lie on the circles in the following order:
A
,
B
,
P
,
Q
A, B, P, Q
A
,
B
,
P
,
Q
on the circle
w
1
{{w} _ {1} }
w
1
and
C
,
D
,
P
,
Q
C, D, P, Q
C
,
D
,
P
,
Q
on the circle
w
2
{{w} _ {2}}
w
2
. The lines
A
P
AP
A
P
and
B
Q
BQ
BQ
intersect at the point
X
X
X
, and the lines
C
P
CP
CP
and
D
Q
DQ
D
Q
intersect at the point
Y
,
X
≠
Y
Y, X \ne Y
Y
,
X
=
Y
. Prove that all lines
X
Y
XY
X
Y
for different diameters
A
B
AB
A
B
and
C
D
CD
C
D
pass through the same point or are all parallel.(Serdyuk Nazar)
2010.10.4
1
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fixed point, fixed length, fixed angle (2010 Kyiv City MO Round2 10.4 11.4)
The points
A
≠
B
A \ne B
A
=
B
are given on the plane. The point
C
C
C
moves along the plane in such a way that
∠
A
C
B
=
α
\angle ACB = \alpha
∠
A
CB
=
α
, where
α
\alpha
α
is the fixed angle from the interval (
0
o
,
18
0
o
0^o, 180^o
0
o
,
18
0
o
). The circle inscribed in triangle
A
B
C
ABC
A
BC
has center the point
I
I
I
and touches the sides
A
B
,
B
C
,
C
A
AB, BC, CA
A
B
,
BC
,
C
A
at points
D
,
E
,
F
D, E, F
D
,
E
,
F
accordingly. Rays
A
I
AI
A
I
and
B
I
BI
B
I
intersect the line
E
F
EF
EF
at points
M
M
M
and
N
N
N
, respectively. Show that: a) the segment
M
N
MN
MN
has a constant length, b) all circles circumscribed around triangle
D
M
N
DMN
D
MN
have a common point
2011.11.4
1
Hide problems
concurrency, // diameters of 3 tangent circles (2011 Kyiv City MO Round2 11.4)
Let three circles be externally tangent in pairs, with parallel diameters
A
1
A
2
,
B
1
B
2
,
C
1
C
2
A_1A_2, B_1B_2, C_1C_2
A
1
A
2
,
B
1
B
2
,
C
1
C
2
(i.e. each of the quadrilaterals
A
1
B
1
B
2
A
2
A_1B_1B_2A_2
A
1
B
1
B
2
A
2
and
A
1
C
1
C
2
A
2
A_1C_1C_2A_2
A
1
C
1
C
2
A
2
is a parallelogram or trapezoid, which segment
A
1
A
2
A_1A_2
A
1
A
2
is the base). Prove that
A
1
B
2
,
B
1
C
2
,
C
1
A
2
A_1B_2, B_1C_2, C_1A_2
A
1
B
2
,
B
1
C
2
,
C
1
A
2
intersect at one point.(Yuri Biletsky )