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Random Geometry Problems from Ukrainian Contests
Ukraine Correspondence MO - geometry
Ukraine Correspondence MO - geometry
Part of
Random Geometry Problems from Ukrainian Contests
Subcontests
(50)
2003.11
1
Hide problems
ВС/US = DE/SТ = FА/ТU if АВ/РR = CD/RQ= ЕF/QP, convex hexagon ABCDEF
Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a convex hexagon,
P
,
Q
,
R
P, Q, R
P
,
Q
,
R
be the intersection points of
A
B
AB
A
B
and
E
F
EF
EF
,
E
F
EF
EF
and
C
D
CD
C
D
,
C
D
CD
C
D
and
A
B
AB
A
B
.
S
,
T
,
U
V
S, T,UV
S
,
T
,
U
V
are the intersection points of
B
C
BC
BC
and
D
E
DE
D
E
,
D
E
DE
D
E
and
F
A
FA
F
A
,
F
A
FA
F
A
and
B
C
BC
BC
, respectively. Prove that if
A
B
P
R
=
C
D
R
Q
=
E
F
Q
P
,
\frac{AB}{PR}=\frac{CD}{RQ}=\frac{EF}{QP},
PR
A
B
=
RQ
C
D
=
QP
EF
,
then
B
C
U
S
=
D
E
S
T
=
F
A
T
U
.
\frac{BC}{US}=\frac{DE}{ST}=\frac{FA}{TU}.
U
S
BC
=
ST
D
E
=
T
U
F
A
.
2003.8
1
Hide problems
angle wanted, CE=BC/3 (2003 VIII All-Ukrainian Correspondence MO 5-11 p8)
In the triangle
A
B
C
ABC
A
BC
,
D
D
D
is the midpoint of
A
B
AB
A
B
, and
E
E
E
is the point on the side
B
C
BC
BC
, for which
C
E
=
1
3
B
C
CE = \frac13 BC
CE
=
3
1
BC
. It is known that
∠
A
D
C
=
∠
B
A
E
\angle ADC =\angle BAE
∠
A
D
C
=
∠
B
A
E
. Find
∠
B
A
C
\angle BAC
∠
B
A
C
.
2003.5
1
Hide problems
max angle <OPA (2003 VIII All-Ukrainian Correspondence MO 5-11 p5)
Let
O
O
O
be the center of the circle
ω
\omega
ω
, and let
A
A
A
be a point inside this circle, different from
O
O
O
. Find all points
P
P
P
on the circle
ω
\omega
ω
for which the angle
∠
O
P
A
\angle OPA
∠
OP
A
acquires the greatest value.
2021.7
1
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DE passes through I, if <BAC = 60 ^o and BD = CE = BC
Let
I
I
I
be the center of a circle inscribed in triangle
A
B
C
ABC
A
BC
, in which
∠
B
A
C
=
6
0
o
\angle BAC = 60 ^o
∠
B
A
C
=
6
0
o
and
A
B
≠
A
C
AB \ne AC
A
B
=
A
C
. The points
D
D
D
and
E
E
E
were marked on the rays
B
A
BA
B
A
and
C
A
CA
C
A
so that
B
D
=
C
E
=
B
C
BD = CE = BC
B
D
=
CE
=
BC
. Prove that the line
D
E
DE
D
E
passes through the point
I
I
I
.
2021.11
1
Hide problems
BC bisects <FBG, if BD = CD and EF //CD
Let
D
D
D
be a point on the side
A
B
AB
A
B
of the triangle
A
B
C
ABC
A
BC
such that
B
D
=
C
D
BD = CD
B
D
=
C
D
, and let the points
E
E
E
on the side
B
C
BC
BC
and
F
F
F
on the extension
A
C
AC
A
C
beyond the point
C
C
C
be such that
E
F
∥
C
D
EF\parallel CD
EF
∥
C
D
. The lines
A
E
AE
A
E
and
C
D
CD
C
D
intersect at the point
G
G
G
. Prove that
B
C
BC
BC
is the bisector of the angle
F
B
G
FBG
FBG
.
2013.12
1
Hide problems
every 3 km turns to the left, every 7 turns to the right
Krut and Vert go by car from point
A
A
A
to point
B
B
B
. The car leaves
A
A
A
in the direction of
B
B
B
, but every
3
3
3
km of the road Krut turns
9
0
o
90^o
9
0
o
to the left, and every
7
7
7
km of the road Vert turns
9
0
o
90^o
9
0
o
to the right ( if they try to turn at the same time, the car continues to go in the same direction). Will Krut and Vert be able to get to
B
B
B
if the distance between
A
A
A
and
B
B
B
is
100
100
100
km?
2019.8
1
Hide problems
ratio wanted, 5 congruent regular pentagons inside a regular pentagon
The symbol of the Olympiad shows
5
5
5
regular hexagons with side
a
a
a
, located inside a regular hexagon with side
b
b
b
. Find ratio
a
b
\frac{a}{b}
b
a
. https://1.bp.blogspot.com/-OwyAl75LwiM/YIsThl3SG6I/AAAAAAAANS0/LwHEsAfyZMcqVIS8h_jr_n46OcMJaSTgQCLcBGAsYHQ/s0/2019%2BUkraine%2Bcorrespondence%2B5-12%2Bp8.png
2020.11
1
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R lies on AC wanted, starting with cyclic ABCD, 2 circumcenters,
The diagonals of the cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
intersect at the point
E
E
E
. Let
P
P
P
and
Q
Q
Q
are the centers of the circles circumscribed around the triangles
B
C
E
BCE
BCE
and
D
C
E
DCE
D
CE
, respectively. A straight line passing through the point
P
P
P
parallel to
A
B
AB
A
B
, and a straight line passing through the point
Q
Q
Q
parallel to
A
D
AD
A
D
, intersect at the point
R
R
R
. Prove that the point
R
R
R
lies on segment
A
C
AC
A
C
.
2020.8
1
Hide problems
EF_|_AD wanted, angle bisectors, circumcircles related
Let
A
B
C
ABC
A
BC
be an acute triangle,
D
D
D
be the midpoint of
B
C
BC
BC
. Bisectors of angles
A
D
B
ADB
A
D
B
and
A
D
C
ADC
A
D
C
intersect the circles circumscribed around the triangles
A
D
B
ADB
A
D
B
and
A
D
C
ADC
A
D
C
at points
E
E
E
and
F
F
F
, respectively. Prove that
E
F
⊥
A
D
EF\perp AD
EF
⊥
A
D
.
2019.11
1
Hide problems
PQ _|_ AC wanted, arc midpoint of circumcircle related
Let
O
O
O
be the center of the circle circumscribed around the acute triangle
A
B
C
ABC
A
BC
, and let
N
N
N
be the midpoint of the arc
A
B
C
ABC
A
BC
of this circle. On the sides
A
B
AB
A
B
and
B
C
BC
BC
mark points
D
D
D
and
E
E
E
respectively, such that the point
O
O
O
lies on the segment
D
E
DE
D
E
. The lines
D
N
DN
D
N
and
B
C
BC
BC
intersect at the point
P
P
P
, and the lines
E
N
EN
EN
and
A
B
AB
A
B
intersect at the point
Q
Q
Q
. Prove that
P
Q
⊥
A
C
PQ \perp AC
PQ
⊥
A
C
.
2019.7
1
Hide problems
point D construction on AB with BD = CE and <ADC =<BEC
Given a triangle
A
B
C
ABC
A
BC
. Construct a point
D
D
D
on the side
A
B
AB
A
B
and point
E
E
E
on the side
A
C
AC
A
C
so that
B
D
=
C
E
BD = CE
B
D
=
CE
and
∠
A
D
C
=
∠
B
E
C
\angle ADC = \angle BEC
∠
A
D
C
=
∠
BEC
2011.3
1
Hide problems
cut a square out of a rectangle (2011 All-Ukrainian Correspondence MO 5-12 p3)
The kid cut out of grid paper with the side of the cell
1
1
1
rectangle along the grid lines and calculated its area and perimeter. Carlson snatched his scissors and cut out of this rectangle along the lines of the grid a square adjacent to the boundary of the rectangle. - My rectangle ... - kid sobbed. - There is something strange about this figure! - Nonsense, do not mention it - Carlson said - waving his hand carelessly. - Here you see, in this figure the perimeter is the same as the area of the rectangle was, and the area is the same as was the perimeter! What size square did Carlson cut out?
2018.9
1
Hide problems
circumcenter of ADE lies on segments AC, orthcenter related
Let
A
B
C
ABC
A
BC
be an acute-angled triangle in which
A
B
<
A
C
AB <AC
A
B
<
A
C
. On the side
B
C
BC
BC
mark a point
D
D
D
such that
A
D
=
A
B
AD = AB
A
D
=
A
B
, and on the side
A
B
AB
A
B
mark a point
E
E
E
such that the segment
D
E
DE
D
E
passes through the orthocenter of triangle
A
B
C
ABC
A
BC
. Prove that the center of the circumcircle of triangle
A
D
E
ADE
A
D
E
lies on the segment
A
C
AC
A
C
.
2018.6
1
Hide problems
AB = 2DE if <B = 2<C where AD , AE altitude ,median of ABC
Let
A
D
AD
A
D
and
A
E
AE
A
E
be the altitude and median of triangle
A
B
C
ABC
A
BC
, in with
∠
B
=
2
∠
C
\angle B = 2\angle C
∠
B
=
2∠
C
. Prove that
A
B
=
2
D
E
AB = 2DE
A
B
=
2
D
E
.
2017.11
1
Hide problems
exists circle tangent to 4 circumcircles, #, <APB+<CPD=<BPC+<APD
Inside the parallelogram
A
B
C
D
ABCD
A
BC
D
, choose a point
P
P
P
such that
∠
A
P
B
+
∠
C
P
D
=
∠
B
P
C
+
∠
A
P
D
\angle APB+ \angle CPD= \angle BPC+ \angle APD
∠
A
PB
+
∠
CP
D
=
∠
BPC
+
∠
A
P
D
. Prove that there exists a circle tangent to each of the circles circumscribed around the triangles
A
P
B
APB
A
PB
,
B
P
C
BPC
BPC
,
C
P
D
CPD
CP
D
and
A
P
D
APD
A
P
D
.
2017.8
1
Hide problems
find point K on midline of isosceles trapezoid with min sum <DAK + <BCK
On the midline of the isosceles trapezoid
A
B
C
D
ABCD
A
BC
D
(
B
C
∥
A
D
BC \parallel AD
BC
∥
A
D
) find the point
K
K
K
, for which the sum of the angles
∠
D
A
K
+
∠
B
C
K
\angle DAK + \angle BCK
∠
D
A
K
+
∠
BC
K
will be the smallest.
2009.11
1
Hide problems
angle wanted, bisector AD=\sqrt{BD xCD}, <ADB = 45^o
In triangle
A
B
C
ABC
A
BC
, the length of the angle bisector
A
D
AD
A
D
is
B
D
⋅
C
D
\sqrt{BD \cdot CD}
B
D
⋅
C
D
. Find the angles of the triangle
A
B
C
ABC
A
BC
, if
∠
A
D
B
=
4
5
o
\angle ADB = 45^o
∠
A
D
B
=
4
5
o
.
2009.7
1
Hide problems
<CBD=<ADF wanted, pentagon, AE//BC, <ADE = <BDC
Let
A
B
C
D
E
ABCDE
A
BC
D
E
be a convex pentagon such that
A
E
∥
B
C
AE\parallel BC
A
E
∥
BC
and
∠
A
D
E
=
∠
B
D
C
\angle ADE = \angle BDC
∠
A
D
E
=
∠
B
D
C
. The diagonals
A
C
AC
A
C
and
B
E
BE
BE
intersect at point
F
F
F
. Prove that
∠
C
B
D
=
∠
A
D
F
\angle CBD= \angle ADF
∠
CB
D
=
∠
A
D
F
.
2009.3
1
Hide problems
inradius of rightangled, compass (2009 All-Ukrainian Correspondence MO 5-12 p3)
A right triangle is drawn on the plane. How to use only a compass to mark two points, such that the distance between them is equal to the diameter of the circle inscribed in this triangle?
2007.11
1
Hide problems
AQ/AP wanted, 2 circumcircles each passing through a midpoint
Denote by
B
1
B_1
B
1
and
C
1
C_1
C
1
, the midpoints of the sides
A
B
AB
A
B
and
A
C
AC
A
C
of the triangle
A
B
C
ABC
A
BC
. Let the circles circumscribed around the triangles
A
B
C
1
ABC_1
A
B
C
1
and
A
B
1
C
AB_1C
A
B
1
C
intersect at points
A
A
A
and
P
P
P
, and let the line
A
P
AP
A
P
intersect the circle circumscribed around the triangle
A
B
C
ABC
A
BC
at points
A
A
A
and
Q
Q
Q
. Find the ratio
A
Q
A
P
\frac{AQ}{AP}
A
P
A
Q
.
2007.9
1
Hide problems
min perimeter, triangle with integer sidelengths, <B = 2 <А, <C > 90^o
In triangle
A
B
C
ABC
A
BC
, the lengths of all sides are integers,
∠
B
=
2
∠
A
\angle B=2 \angle A
∠
B
=
2∠
A
and
∠
C
>
9
0
o
\angle C> 90^o
∠
C
>
9
0
o
. Find the smallest possible perimeter of this triangle.
2007.7
1
Hide problems
<ANC = ? isosceles ABC, <BND = 90^o, midpoints
Let
A
B
C
ABC
A
BC
be an isosceles triangle (
A
B
=
A
C
AB = AC
A
B
=
A
C
),
D
D
D
be the midpoint of
B
C
BC
BC
, and
M
M
M
be the midpoint of
A
D
AD
A
D
. On the segment
B
M
BM
BM
take a point
N
N
N
such that
∠
B
N
D
=
9
0
o
\angle BND = 90^o
∠
BN
D
=
9
0
o
. Find the angle
A
N
C
ANC
A
NC
.
2004.6
1
Hide problems
_|_ construction by ruler (2004 All-Ukrainian Correspondence MO 5-11 p6)
A circle is drawn on the plane. How to use only a ruler to draw a perpendicular from a given point outside the circle to a given line passing through the center of this circle?
2006.3
1
Hide problems
locus of orthocenters inscribed in given circle
Find the locus of the points of intersection of the altitudes of the triangles inscribed in a given circle.
2006.7
1
Hide problems
5/6 <= AD/AB <= \sqrt{73}/10 , right triangle, <CAD=<ABE
Let
D
D
D
and
E
E
E
be the midpoints of the sides
B
C
BC
BC
and
A
C
AC
A
C
of a right triangle
A
B
C
ABC
A
BC
. Prove that if
∠
C
A
D
=
∠
A
B
E
\angle CAD=\angle ABE
∠
C
A
D
=
∠
A
BE
, then
5
6
≤
A
D
A
B
≤
73
10
.
\frac{5}{6} \le \frac{AD}{AB}\le \frac{\sqrt{73}}{10}.
6
5
≤
A
B
A
D
≤
10
73
.
2006.10
1
Hide problems
sum of inradius + radius of mixtiinear excircle is fixed, start with isosceles
Let
A
B
C
ABC
A
BC
be an isosceles triangle (
A
B
=
A
C
AB=AC
A
B
=
A
C
). An arbitrary point
M
M
M
is chosen on the extension of the
B
C
BC
BC
beyond point
B
B
B
. Prove that the sum of the radius of the circle inscribed in the triangle
A
M
B
AMB
A
M
B
and the radius of the circle tangent to the side
A
C
AC
A
C
and the extensions of the sides
A
M
,
C
M
AM, CM
A
M
,
CM
of the triangle
A
M
C
AMC
A
MC
does not depend on the choice of point
M
M
M
.
2005.4
1
Hide problems
<C wanted, AE + BD = AB, angle bisectors
The bisectors of the angles
A
A
A
and
B
B
B
of the triangle
A
B
C
ABC
A
BC
intersect the sides
B
C
BC
BC
and
A
C
AC
A
C
at points
D
D
D
and
E
E
E
. It is known that
A
E
+
B
D
=
A
B
AE + BD = AB
A
E
+
B
D
=
A
B
. Find the angle
∠
C
\angle C
∠
C
.
2005.7
1
Hide problems
<AOE =? trapezoid ABCD, EF//AB , EF = AD, <AOB=<DAB=90^o
Let
O
O
O
be the point of intersection of the diagonals of the trapezoid
A
B
C
D
ABCD
A
BC
D
with the bases
A
B
AB
A
B
and
C
D
CD
C
D
. It is known that
∠
A
O
B
=
∠
D
A
B
=
9
0
o
\angle AOB = \angle DAB = 90^o
∠
A
OB
=
∠
D
A
B
=
9
0
o
. On the sides
A
D
AD
A
D
and
B
C
BC
BC
take the points
E
E
E
and
F
F
F
so that
E
F
∥
A
B
EF\parallel AB
EF
∥
A
B
and
E
F
=
A
D
EF = AD
EF
=
A
D
. Find the angle
∠
A
O
E
\angle AOE
∠
A
OE
.
2005.11
1
Hide problems
tangent wanted, sstrarin with circumcircle
Let the circle
ω
\omega
ω
be circumscribed around the triangle
△
A
B
C
\vartriangle ABC
△
A
BC
with right angle
∠
A
\angle A
∠
A
. Tangent to the circle
ω
\omega
ω
at point
A
A
A
intersects the line
B
C
BC
BC
at point
D
D
D
. Point
E
E
E
is symmetric to
A
A
A
with respect to the line
B
C
BC
BC
. Let
K
K
K
be the foot of the perpendicular drawn from point
A
A
A
on
B
E
BE
BE
,
L
L
L
the midpoint of
A
K
AK
A
K
. The line
B
L
BL
B
L
intersects the circle
ω
\omega
ω
for the second time at the point
N
N
N
. Prove that the line
B
D
BD
B
D
is tangent to the circle circumscribed around the triangle
△
A
D
M
\vartriangle ADM
△
A
D
M
.
2008.11
1
Hide problems
concyclic wanted, starting with a #, a circle and a perpendicular
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram. A circle with diameter
A
C
AC
A
C
intersects line
B
D
BD
B
D
at points
P
P
P
and
Q
Q
Q
. The perpendicular on
A
C
AC
A
C
passing through point
C
C
C
, intersects lines
A
B
AB
A
B
and
A
D
AD
A
D
at points
X
X
X
and
Y
Y
Y
, respectively. Prove that the points
P
,
Q
,
X
P, Q, X
P
,
Q
,
X
and
Y
Y
Y
lie on the same circle.
2008.7
1
Hide problems
angles wanted, <ABD = <CBD, 3<ACE = 2<BCE, CD = DE = CH
On the sides
A
C
AC
A
C
and
A
B
AB
A
B
of the triangle
A
B
C
ABC
A
BC
, the points
D
D
D
and
E
E
E
were chosen such that
∠
A
B
D
=
∠
C
B
D
\angle ABD =\angle CBD
∠
A
B
D
=
∠
CB
D
and
3
∠
A
C
E
=
2
∠
B
C
E
3 \angle ACE = 2\angle BCE
3∠
A
CE
=
2∠
BCE
. Let
H
H
H
be the point of intersection of
B
D
BD
B
D
and
C
E
CE
CE
, and
C
D
=
D
E
=
C
H
CD = DE = CH
C
D
=
D
E
=
C
H
. Find the angles of triangle
A
B
C
ABC
A
BC
.
2004.8
1
Hide problems
ratio of areas AEGH/ABCD in trapezoid ABCD, midpoints
The extensions of the sides
A
B
AB
A
B
and
C
D
CD
C
D
of the trapezoid
A
B
C
D
ABCD
A
BC
D
intersect at point
E
E
E
. Denote by
H
H
H
and
G
G
G
the midpoints of
B
D
BD
B
D
and
A
C
AC
A
C
. Find the ratio of the area
A
E
G
H
AEGH
A
EG
H
to the area
A
B
C
D
ABCD
A
BC
D
.
2004.10.
1
Hide problems
<A=? if BC = BD + AD in isosceles ABC, angle bisector
In an isosceles triangle
A
B
C
ABC
A
BC
(
A
B
=
A
C
AB = AC
A
B
=
A
C
), the bisector of the angle
B
B
B
intersects
A
C
AC
A
C
at point
D
D
D
such that
B
C
=
B
D
+
A
D
BC = BD + AD
BC
=
B
D
+
A
D
. Find
∠
A
\angle A
∠
A
.
2016.11
1
Hide problems
angles wanted, isosceles trapezoid, inside a square, <BAP=30^o, <BCP=15^o
Inside the square
A
B
C
D
ABCD
A
BC
D
mark the point
P
P
P
, for which
∠
B
A
P
=
3
0
o
\angle BAP = 30^o
∠
B
A
P
=
3
0
o
and
∠
B
C
P
=
1
5
o
\angle BCP = 15^o
∠
BCP
=
1
5
o
. The point
Q
Q
Q
was chosen so that
A
P
C
Q
APCQ
A
PCQ
is an isosceles trapezoid (
P
C
∥
A
Q
PC\parallel AQ
PC
∥
A
Q
). Find the angles of the triangle
C
A
M
CAM
C
A
M
, where
M
M
M
is the midpoint of
P
Q
PQ
PQ
.
2016.7
1
Hide problems
collinear wanted, starting with incircle of isosceles triangle
The circle
ω
\omega
ω
inscribed in an isosceles triangle
A
B
C
ABC
A
BC
(
A
C
=
B
C
AC = BC
A
C
=
BC
) touches the side
B
C
BC
BC
at point
D
D
D
.On the extensions of the segment
A
B
AB
A
B
beyond points
A
A
A
and
B
B
B
, respectively mark the points
K
K
K
and
L
L
L
so that
A
K
=
B
L
AK = BL
A
K
=
B
L
, The lines
K
D
KD
KD
and
L
D
LD
L
D
intersect the circle
ω
\omega
ω
for second time at points
G
G
G
and
H
H
H
, respectively. Prove that point
A
A
A
belongs to the line
G
H
GH
G
H
.
2015.11
1
Hide problems
collinear wanted, 6 circumcircles related, isosceles, altitudes, midpoints
Let
A
B
C
ABC
A
BC
be an non- isosceles triangle,
H
a
H_a
H
a
,
H
b
H_b
H
b
, and
H
c
H_c
H
c
be the feet of the altitudes drawn from the vertices
A
,
B
A, B
A
,
B
, and
C
C
C
, respectively, and
M
a
M_a
M
a
,
M
b
M_b
M
b
, and
M
c
M_c
M
c
be the midpoints of the sides
B
C
BC
BC
,
C
A
CA
C
A
, and
A
B
AB
A
B
, respectively. The circumscribed circles of triangles
A
H
b
H
c
AH_bH_c
A
H
b
H
c
and
A
M
b
M
c
AM_bM_c
A
M
b
M
c
intersect for second time at point
A
′
A'
A
′
. The circumscribed circles of triangles
B
H
c
H
a
BH_cH_a
B
H
c
H
a
and
B
M
c
M
a
BM_cM_a
B
M
c
M
a
intersect for second time at point
B
′
B'
B
′
. The circumscribed circles of triangles
C
H
a
H
b
CH_aH_b
C
H
a
H
b
and
C
M
a
M
b
CM_aM_b
C
M
a
M
b
intersect for second time at point
C
′
C'
C
′
. Prove that points
A
′
,
B
′
A', B'
A
′
,
B
′
and
C
′
C'
C
′
lie on the same line.
2015.8
1
Hide problems
DA bisector of <EDF if <AEF = <FDB, <AFE = <EDC, equilateral ABC
On the sides
B
C
,
A
C
BC, AC
BC
,
A
C
and
A
B
AB
A
B
of the equilateral triangle
A
B
C
ABC
A
BC
mark the points
D
,
E
D, E
D
,
E
and
F
F
F
so that
∠
A
E
F
=
∠
F
D
B
\angle AEF = \angle FDB
∠
A
EF
=
∠
F
D
B
and
∠
A
F
E
=
∠
E
D
C
\angle AFE = \angle EDC
∠
A
FE
=
∠
E
D
C
. Prove that
D
A
DA
D
A
is the bisector of the angle
E
D
F
EDF
E
D
F
.
2014.12
1
Hide problems
equal angles wanted, mixtilinear excircle related
Let
ω
\omega
ω
be the circumscribed circle of triangle
A
B
C
ABC
A
BC
, and let
ω
′
\omega'
ω
′
'be the circle tangent to the side
B
C
BC
BC
and the extensions of the sides
A
B
AB
A
B
and
A
C
AC
A
C
. The common tangents to the circles
ω
\omega
ω
and
ω
′
\omega'
ω
′
intersect the line
B
C
BC
BC
at points
D
D
D
and
E
E
E
. Prove that
∠
B
A
D
=
∠
C
A
E
\angle BAD = \angle CAE
∠
B
A
D
=
∠
C
A
E
.
2014.10
1
Hide problems
one of lines DE,DF passes through incenter of ABC, circumcircle related
In the triangle
A
B
C
ABC
A
BC
, it is known that
A
C
<
A
B
AC <AB
A
C
<
A
B
. Let
ℓ
\ell
ℓ
be tangent to the circumcircle of triangle
A
B
C
ABC
A
BC
drawn at point
A
A
A
. A circle with center
A
A
A
and radius
A
C
AC
A
C
intersects segment
A
B
AB
A
B
at point
D
D
D
, and line
ℓ
\ell
ℓ
at points
E
E
E
and
F
F
F
. Prove that one of the lines
D
E
DE
D
E
and
D
F
DF
D
F
passes through the center inscribed circle of triangle
A
B
C
ABC
A
BC
.
2014.7
1
Hide problems
BC bisector of <DBE wanted, AB=AC, AC = 2AD and AE = 2AC
Let
A
B
C
ABC
A
BC
be an isosceles triangle (
A
B
=
A
C
AB = AC
A
B
=
A
C
). The points
D
D
D
and
E
E
E
were marked on the ray
A
C
AC
A
C
so that
A
C
=
2
A
D
AC = 2AD
A
C
=
2
A
D
and
A
E
=
2
A
C
AE = 2AC
A
E
=
2
A
C
. Prove that
B
C
BC
BC
is the bisector of the angle
∠
D
B
E
\angle DBE
∠
D
BE
.
2013.11
1
Hide problems
midpoint wanted, 2 circles realted
Given a triangle
A
B
C
ABC
A
BC
. The circle
ω
1
\omega_1
ω
1
passes through the vertex
B
B
B
and touches the side
A
C
AC
A
C
at the point
A
A
A
, and the circle
ω
2
\omega_2
ω
2
passes through the vertex
C
C
C
and touches the side
A
B
AB
A
B
at the point
A
A
A
. The circles
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
intersect a second time at the point
D
D
D
. The line
A
D
AD
A
D
intersects the circumcircle of the triangle
A
B
C
ABC
A
BC
at point
E
E
E
. Prove that
D
D
D
is the midpoint of
A
E
AE
A
E
..
2013.9
1
Hide problems
2 equal circumcircles wanted, midpoints of cyclic ABCD related
Let
E
E
E
be the point of intersection of the diagonals of the cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
, and let
K
,
L
,
M
K, L, M
K
,
L
,
M
and
N
N
N
be the midpoints of the sides
A
B
,
B
C
,
C
D
AB, BC, CD
A
B
,
BC
,
C
D
and
D
A
DA
D
A
, respectively. Prove that the radii of the circles circumscribed around the triangles
K
L
E
KLE
K
L
E
and
M
N
E
MNE
MNE
are equal.
2013.7
1
Hide problems
collinear wanted, right triangle and 2 circles
An arbitrary point
D
D
D
is marked on the hypotenuse
A
B
AB
A
B
of a right triangle
A
B
C
ABC
A
BC
. The circle circumscribed around the triangle
A
C
D
ACD
A
C
D
intersects the line
B
C
BC
BC
at the point
E
E
E
for the second time, and the circle circumscribed around the triangle
B
C
D
BCD
BC
D
intersects the line
A
C
AC
A
C
for the second time at the point
F
F
F
. Prove that the line
E
F
EF
EF
passes through the point
D
D
D
.
2012.10
1
Hide problems
equal sums of 2 sides, cyclic ABCD, <BAD = 60^o, AO = 3OC
The diagonals
A
C
AC
A
C
and
B
D
BD
B
D
of the cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
intersect at a point O. It is known that
∠
B
A
D
=
6
0
o
\angle BAD = 60^o
∠
B
A
D
=
6
0
o
and
A
O
=
3
O
C
AO = 3OC
A
O
=
3
OC
. Prove that the sum of some two sides of a quadrilateral is equal to the sum of the other two sides.
2012.7
1
Hide problems
circumcircle of ADE bisects OH, circumcenter, orthocenter
Let
O
O
O
and
H
H
H
be the center of the circumcircle and the point of intersection of the altitudes of the acute triangle
A
B
C
ABC
A
BC
respectively,
D
D
D
be the foot of the altitude drawn to
B
C
BC
BC
, and
E
E
E
be the midpoint of
A
O
AO
A
O
. Prove that the circumcircle of the triangle
A
D
E
ADE
A
D
E
passes through the midpoint of the segment
O
H
OH
O
H
.
2011.11
1
Hide problems
cyclic wanted, points symmetric to intersection of perp. diagonals wrt sides
In a quadrilateral
A
B
C
D
ABCD
A
BC
D
, the diagonals are perpendicular and intersect at the point
S
S
S
. Let
K
,
L
,
M
K, L, M
K
,
L
,
M
, and
N
N
N
be points symmetric to
S
S
S
with respect to the lines
A
B
,
B
C
,
C
D
AB, BC, CD
A
B
,
BC
,
C
D
, and
D
A
DA
D
A
, respectively,
B
N
BN
BN
intersects the circumcircle of the triangle
S
K
N
SKN
S
K
N
at point
E
E
E
, and
B
M
BM
BM
intersects circumscribed the circle of the triangle
S
L
M
SLM
S
L
M
at the point
F
F
F
. Prove that the quadrilateral
E
F
L
K
EFLK
EF
L
K
is cyclic .
2011.9
1
Hide problems
AM = CN = a =? if B,M, N are collinear, regular hexagon
On the diagonals
A
C
AC
A
C
and
C
E
CE
CE
of a regular hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
with side
1
1
1
we mark points
M
M
M
and
N
N
N
such that
A
M
=
C
N
=
a
AM = CN = a
A
M
=
CN
=
a
. Find
a
a
a
if the points
B
,
M
,
N
B, M, N
B
,
M
,
N
lie on the same line.
2011.7
1
Hide problems
AP _|_ BQ wanted, trapezoid with bases AB = 2CD, circle
Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid in which
A
B
∥
C
D
AB \parallel CD
A
B
∥
C
D
and
A
B
=
2
C
D
AB = 2CD
A
B
=
2
C
D
. A line
ℓ
\ell
ℓ
perpendicular to
C
D
CD
C
D
was drawn through point
C
C
C
. A circle with center at point
D
D
D
and radius
D
A
DA
D
A
intersects line
ℓ
\ell
ℓ
at points
P
P
P
and
Q
Q
Q
. Prove that
A
P
⊥
B
Q
AP \perp BQ
A
P
⊥
BQ
.
2010.11
1
Hide problems
ratio <ABC: <AHI, <BAC = 60^o, H orthocenter, I incenter
Let
A
B
C
ABC
A
BC
be an acute-angled triangle in which
∠
B
A
C
=
6
0
o
\angle BAC = 60^o
∠
B
A
C
=
6
0
o
and
A
B
>
A
C
AB> AC
A
B
>
A
C
. Let
H
H
H
and
I
I
I
denote the points of intersection of the altitudes and angle bisectors of this triangle, respectively. Find the ratio
∠
A
B
C
:
∠
A
H
I
\angle ABC: \angle AHI
∠
A
BC
:
∠
A
H
I
.
2010.7
1
Hide problems
BE = AD if DE// AB , CE // BM, D in median BM
An arbitrary point
D
D
D
was marked on the median
B
M
BM
BM
of the triangle
A
B
C
ABC
A
BC
. It is known that the point
D
E
∥
A
B
DE\parallel AB
D
E
∥
A
B
and
C
E
∥
B
M
CE \parallel BM
CE
∥
BM
. Prove that
B
E
=
A
D
BE = AD
BE
=
A
D