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National and Regional Contests
Ukraine Contests
Random Geometry Problems from Ukrainian Contests
Champions Tournament Seniors - geometry
Champions Tournament Seniors - geometry
Part of
Random Geometry Problems from Ukrainian Contests
Subcontests
(21)
2011.2
1
Hide problems
<AEF = 90^o if AB=AC, PQ// AB, F circumcener of PQC
Let
A
B
C
ABC
A
BC
be an isosceles triangle in which
A
B
=
A
C
AB = AC
A
B
=
A
C
. On its sides
B
C
BC
BC
and
A
C
AC
A
C
respectively are marked points
P
P
P
and
Q
Q
Q
so that
P
Q
∥
A
B
PQ\parallel AB
PQ
∥
A
B
. Let
F
F
F
be the center of the circle circumscribed about the triangle
P
Q
C
PQC
PQC
, and
E
E
E
the midpoint of the segment
B
Q
BQ
BQ
. Prove that
∠
A
E
F
=
9
0
o
\angle AEF = 90^o
∠
A
EF
=
9
0
o
.
2011.4
1
Hide problems
volume of common part 2 regular quadrangular pyramids
The height
S
O
SO
SO
of a regular quadrangular pyramid
S
A
B
C
D
SABCD
S
A
BC
D
forms an angle
6
0
o
60^o
6
0
o
with a side edge , the volume of this pyramid is equal to
18
18
18
cm
3
^3
3
. The vertex of the second regular quadrangular pyramid is at point
S
S
S
, the center of the base is at point
C
C
C
, and one of the vertices of the base lies on the line
S
O
SO
SO
. Find the volume of the common part of these pyramids. (The common part of the pyramids is the set of all such points in space that lie inside or on the surface of both pyramids).
2001.4
1
Hide problems
BQ = PE in pentagon <ABC =<AED = 90^o, < BAC< DAE
Given a convex pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
in which
∠
A
B
C
=
∠
A
E
D
=
9
0
o
\angle ABC = \angle AED = 90^o
∠
A
BC
=
∠
A
E
D
=
9
0
o
,
∠
B
A
C
=
∠
D
A
E
\angle BAC= \angle DAE
∠
B
A
C
=
∠
D
A
E
. Let
K
K
K
be the midpoint of the side
C
D
CD
C
D
, and
P
P
P
the intersection point of lines
A
D
AD
A
D
and
B
K
BK
B
K
,
Q
Q
Q
be the intersection point of lines
A
C
AC
A
C
and
E
K
EK
E
K
. Prove that
B
Q
=
P
E
BQ = PE
BQ
=
PE
.
2008.4
1
Hide problems
< APB + < CPD = 180^o , sphere inscirbed in pyramid with convex ABCD as base
Given a quadrangular pyramid
S
A
B
C
D
SABCD
S
A
BC
D
, the basis of which is a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
. It is known that the pyramid can be tangent to a sphere. Let
P
P
P
be the point of contact of this sphere with the base
A
B
C
D
ABCD
A
BC
D
. Prove that
∠
A
P
B
+
∠
C
P
D
=
18
0
o
\angle APB + \angle CPD = 180^o
∠
A
PB
+
∠
CP
D
=
18
0
o
.
2008.2
1
Hide problems
R is incenter of CST wanted, right triangle, point symmetric wrt side
Given a right triangle
A
B
C
ABC
A
BC
with
∠
C
=
9
0
o
\angle C=90^o
∠
C
=
9
0
o
. On its hypotenuse
A
B
AB
A
B
is arbitrary mark the point
P
P
P
. The point
Q
Q
Q
is symmetric to the point
P
P
P
wrt
A
C
AC
A
C
. Let the lines
P
Q
PQ
PQ
and
B
Q
BQ
BQ
intersect
A
C
AC
A
C
at points
O
O
O
and
R
R
R
respectively. Denote by
S
S
S
the foot of the perpendicular from the point
R
R
R
on the line
A
B
AB
A
B
(
S
≠
P
S \ne P
S
=
P
), and let
T
T
T
be the intersection point of lines
O
S
OS
OS
and
B
R
BR
BR
. Prove that
R
R
R
is the center of the circle inscribed in the triangle
C
S
T
CST
CST
.
2007.5
1
Hide problems
sum of squares of areas of the black faces equals to white ones, polyhedron
The polyhedron
P
A
B
C
D
Q
PABCDQ
P
A
BC
D
Q
has the form shown in the figure. It is known that
A
B
C
D
ABCD
A
BC
D
is parallelogram, the planes of the triangles of the
P
A
C
PAC
P
A
C
and
P
B
D
PBD
PB
D
mutually perpendicular, and also mutually perpendicular are the planes of triangles
Q
A
C
QAC
Q
A
C
and
Q
B
C
QBC
QBC
. Each face of this polyhedron is painted black or white so that the faces that have a common edge are painted in different colors. Prove that the sum of the squares of the areas of the black faces is equal to the sum of the squares of the areas of the white faces. https://1.bp.blogspot.com/-UM5PKEGGWqc/X1V2cXAFmwI/AAAAAAAAMdw/V-Qr94tZmqkj3_q-5mkSICGF1tMu-b_VwCLcBGAsYHQ/s0/2007.5%2Bchampions%2Btourn.png
2007.3
1
Hide problems
locus of circumcenters as M in BA, N in AC with BM=CN
Given a triangle
A
B
C
ABC
A
BC
. Point
M
M
M
moves along the side
B
A
BA
B
A
and point
N
N
N
moves along the side
A
C
AC
A
C
beyond point
C
C
C
such that
B
M
=
C
N
BM=CN
BM
=
CN
. Find the geometric locus of the centers of the circles circumscribed around the triangle
A
M
N
AMN
A
MN
.
2006.3
1
Hide problems
<BPD = 2 <CPD wanted, <ВАС =< ВРВ, isosceles AB=AC, BD:DC = 2: 1
Let
A
B
C
ABC
A
BC
be an isosceles triangle with
A
B
=
A
C
AB = AC
A
B
=
A
C
. Let
D
D
D
be a point on the base
B
C
BC
BC
such that
B
D
:
D
C
=
2
:
1
BD:DC = 2: 1
B
D
:
D
C
=
2
:
1
. Note on the segment
A
D
AD
A
D
a point
P
P
P
such that
∠
B
A
C
=
∠
B
P
D
\angle BAC= \angle BPD
∠
B
A
C
=
∠
BP
D
. Prove that
∠
B
P
D
=
2
∠
C
P
D
\angle BPD = 2 \angle CPD
∠
BP
D
=
2∠
CP
D
.
2004.2
1
Hide problems
concurrent wanted, intersecting circles related
Two different circles
ω
1
\omega_1
ω
1
,
ω
2
\omega_2
ω
2
, with centers
O
1
,
O
2
O_1, O_2
O
1
,
O
2
respectively intersect at the points
A
,
B
A, B
A
,
B
. The line
O
1
B
O_1B
O
1
B
intersects
ω
2
\omega_2
ω
2
at the point
F
(
F
≠
B
)
F (F \ne B)
F
(
F
=
B
)
, and the line
O
2
B
O_2B
O
2
B
intersects
ω
1
\omega_1
ω
1
at the point
E
(
E
≠
B
)
E (E\ne B)
E
(
E
=
B
)
. A line was drawn through the point
B
B
B
, parallel to the
E
F
EF
EF
, which intersects
ω
1
\omega_1
ω
1
at the point
M
(
M
≠
B
)
M (M \ne B)
M
(
M
=
B
)
, and
ω
2
\omega_2
ω
2
at the point
N
(
N
≠
B
)
N (N\ne B)
N
(
N
=
B
)
. Prove that the lines
M
E
,
A
B
ME, AB
ME
,
A
B
and
N
F
NF
NF
intersect at one point.
2002.2
1
Hide problems
perpendicular wanted, tangent and secant to a circle related
The point
P
P
P
is outside the circle
ω
\omega
ω
with center
O
O
O
. Lines
ℓ
1
\ell_1
ℓ
1
and
ℓ
2
\ell_2
ℓ
2
pass through a point
P
P
P
,
ℓ
1
\ell_1
ℓ
1
touches the circle
ω
\omega
ω
at the point
A
A
A
and
ℓ
2
\ell_2
ℓ
2
intersects
ω
\omega
ω
at the points
B
B
B
and
C
C
C
. Tangent to the circle
ω
\omega
ω
at points
B
B
B
and
C
C
C
intersect at point
Q
Q
Q
. Let
K
K
K
be the point of intersection of the lines
B
C
BC
BC
and
A
Q
AQ
A
Q
. Prove that
(
O
K
)
⊥
(
P
Q
)
(OK) \perp (PQ)
(
O
K
)
⊥
(
PQ
)
.
2003.1
1
Hide problems
concurrent wanted, AD _|_AP, projections on angle bisector
Consider the triangle
A
B
C
ABC
A
BC
, in which
A
B
>
A
C
AB > AC
A
B
>
A
C
. Let
P
P
P
and
Q
Q
Q
be the feet of the perpendiculars dropped from the vertices
B
B
B
and
C
C
C
on the bisector of the angle
B
A
C
BAC
B
A
C
, respectively. On the line
B
C
BC
BC
note point
B
B
B
such that
A
D
⊥
A
P
.
AD \perp AP.
A
D
⊥
A
P
.
Prove that the lines
B
Q
,
P
C
BQ, PC
BQ
,
PC
and
A
D
AD
A
D
intersect at one point.
2000.4
1
Hide problems
AG + BG + CG <= A_1C + B_1C + C_1C, centroid , circumcircle
Let
G
G
G
be the point of intersection of the medians in the triangle
A
B
C
ABC
A
BC
. Let us denote
A
1
,
B
1
,
C
1
A_1, B_1, C_1
A
1
,
B
1
,
C
1
the second points of intersection of lines
A
G
,
B
G
,
C
G
AG, BG, CG
A
G
,
BG
,
CG
with the circle circumscribed around the triangle. Prove that
A
G
+
B
G
+
C
G
≤
A
1
C
+
B
1
C
+
C
1
C
AG + BG + CG \le A_1C + B_1C + C_1C
A
G
+
BG
+
CG
≤
A
1
C
+
B
1
C
+
C
1
C
. (Yasinsky V.A.)
2010.3
1
Hide problems
collinear wanted, 2 parallelograms related
On the sides
A
B
AB
A
B
and
B
C
BC
BC
arbitrarily mark points
M
M
M
and
N
N
N
, respectively. Let
P
P
P
be the point of intersection of segments
A
N
AN
A
N
and
B
M
BM
BM
. In addition, we note the points
Q
Q
Q
and
R
R
R
such that quadrilaterals
M
C
N
Q
MCNQ
MCNQ
and
A
C
B
R
ACBR
A
CBR
are parallelograms. Prove that the points
P
,
Q
P,Q
P
,
Q
and
R
R
R
lie on one line.
2005.2
1
Hide problems
<BLC=2<BAC wanted, symmedian and circumcircle related
Given a triangle
A
B
C
ABC
A
BC
, the line passing through the vertex
A
A
A
symmetric to the median
A
M
AM
A
M
wrt the line containing the bisector of the angle
∠
B
A
C
\angle BAC
∠
B
A
C
intersects the circle circumscribed around the triangle
A
B
C
ABC
A
BC
at points
A
A
A
and
K
K
K
. Let
L
L
L
be the midpoint of the segment
A
K
AK
A
K
. Prove that
∠
B
L
C
=
2
∠
B
A
C
\angle BLC=2\angle BAC
∠
B
L
C
=
2∠
B
A
C
.
2013.3
1
Hide problems
\sqrt{MA_1}+ \sqrt{MB_1}+ \sqrt{MC_1}<= \sqrt{SA+SB+SC}, 3d geo ineq
On the base of the
A
B
C
ABC
A
BC
of the triangular pyramid
S
A
B
C
SABC
S
A
BC
mark the point
M
M
M
and through it were drawn lines parallel to the edges
S
A
,
S
B
SA, SB
S
A
,
SB
and
S
C
SC
SC
, which intersect the side faces at the points
A
1
,
B
1
A1_, B_1
A
1
,
B
1
and
C
1
C_1
C
1
, respectively. Prove that
M
A
1
+
M
B
1
+
M
C
1
≤
S
A
+
S
B
+
S
C
\sqrt{MA_1}+ \sqrt{MB_1}+ \sqrt{MC_1}\le \sqrt{SA+SB+SC}
M
A
1
+
M
B
1
+
M
C
1
≤
S
A
+
SB
+
SC
2012.2
1
Hide problems
collinear circumcenters wanted, <AMC = < BAC = < ACK, AM median
About the triangle
A
B
C
ABC
A
BC
it is known that
A
M
AM
A
M
is its median, and
∠
A
M
C
=
∠
B
A
C
\angle AMC = \angle BAC
∠
A
MC
=
∠
B
A
C
. On the ray
A
M
AM
A
M
lies the point
K
K
K
such that
∠
A
C
K
=
∠
B
A
C
\angle ACK = \angle BAC
∠
A
C
K
=
∠
B
A
C
. Prove that the centers of the circumcircles of the triangles
A
B
C
,
A
B
M
ABC, ABM
A
BC
,
A
BM
and
K
C
M
KCM
K
CM
lie on the same line.
2015.3
1
Hide problems
parallel wanted, incircle, circumcircle and mixtlinear related
Given a triangle
A
B
C
ABC
A
BC
. Let
Ω
\Omega
Ω
be the circumscribed circle of this triangle, and
ω
\omega
ω
be the inscribed circle of this triangle. Let
δ
\delta
δ
be a circle that touches the sides
A
B
AB
A
B
and
A
C
AC
A
C
, and also touches the circle
Ω
\Omega
Ω
internally at point
D
D
D
. The line
A
D
AD
A
D
intersects the circle
Ω
\Omega
Ω
at two points
P
P
P
and
Q
Q
Q
(
P
P
P
lies between
A
A
A
and
Q
Q
Q
). Let
O
O
O
and
I
I
I
be the centers of the circles
Ω
\Omega
Ω
and
ω
\omega
ω
. Prove that
O
D
∥
I
Q
OD \parallel IQ
O
D
∥
I
Q
.
2016.3
1
Hide problems
BD=CD+AE wanted, equilateral and parallel to side related
Let
t
t
t
be a line passing through the vertex
A
A
A
of the equilateral
A
B
C
ABC
A
BC
, parallel to the side
B
C
BC
BC
. On the side
A
C
AC
A
C
arbitrarily mark the point
D
D
D
. Bisector of the angle
A
B
D
ABD
A
B
D
intersects the line
t
t
t
at the point
E
E
E
. Prove that
B
D
=
C
D
+
A
E
BD=CD+AE
B
D
=
C
D
+
A
E
.
2017.4
1
Hide problems
KL=BC /2 wanted, circle passing through A tangent to BC at D, angle bisector
Let
A
D
AD
A
D
be the bisector of triangle
A
B
C
ABC
A
BC
. Circle
ω
\omega
ω
passes through the vertex
A
A
A
and touches the side
B
C
BC
BC
at point
D
D
D
. This circle intersects the sides
A
C
AC
A
C
and
A
B
AB
A
B
for the second time at points
M
M
M
and
N
N
N
respectively. Lines
B
M
BM
BM
and
C
N
CN
CN
intersect the circle for the second time
ω
\omega
ω
at points
P
P
P
and
Q
Q
Q
, respectively. Lines
A
P
AP
A
P
and
A
Q
AQ
A
Q
intersect side
B
C
BC
BC
at points
K
K
K
and
L
L
L
, respectively. Prove that
K
L
=
1
2
B
C
KL=\frac12 BC
K
L
=
2
1
BC
2018.3
1
Hide problems
<CDE=<BEF wanted, 2 parallelograms, AC=AD, AE=2CD
The vertex
F
F
F
of the parallelogram
A
C
E
F
ACEF
A
CEF
lies on the side
B
C
BC
BC
of parallelogram
A
B
C
D
ABCD
A
BC
D
. It is known that
A
C
=
A
D
AC = AD
A
C
=
A
D
and
A
E
=
2
C
D
AE = 2CD
A
E
=
2
C
D
. Prove that
∠
C
D
E
=
∠
B
E
F
\angle CDE = \angle BEF
∠
C
D
E
=
∠
BEF
.
2019.2
1
Hide problems
perpendicular wanted, cyclic ABCD, BC=DFC, AB=AC, arc midpoint
The quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed in the circle and the lengths of the sides
B
C
BC
BC
and
D
C
DC
D
C
are equal, and the length of the side
A
B
AB
A
B
is equal to the length of the diagonal
A
C
AC
A
C
. Let the point
P
P
P
be the midpoint of the arc
C
D
CD
C
D
, which does not contain point
A
A
A
, and
Q
Q
Q
is the point of intersection of diagonals
A
C
AC
A
C
and
B
D
BD
B
D
. Prove that the lines
P
Q
PQ
PQ
and
A
B
AB
A
B
are perpendicular.