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Problems
Contests
National and Regional Contests
Russia Contests
Sharygin Geometry Olympiad
2015 Sharygin Geometry Olympiad
2015 Sharygin Geometry Olympiad
Part of
Sharygin Geometry Olympiad
Subcontests
(32)
P24
1
Hide problems
4 lines concurrent, in an interior point, starting with insphere of tetrahedron
The insphere of a tetrahedron ABCD with center
O
O
O
touches its faces at points
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
and
D
1
D_1
D
1
. a) Let
P
a
P_a
P
a
be a point such that its reflections in lines
O
B
,
O
C
OB,OC
OB
,
OC
and
O
D
OD
O
D
lie on plane
B
C
D
BCD
BC
D
. Points
P
b
,
P
c
P_b, P_c
P
b
,
P
c
and
P
d
P_d
P
d
are defined similarly. Prove that lines
A
1
P
a
,
B
1
P
b
,
C
1
P
c
A_1P_a,B_1P_b,C_1P_c
A
1
P
a
,
B
1
P
b
,
C
1
P
c
and
D
1
P
d
D_1P_d
D
1
P
d
concur at some point
P
P
P
. b) Let
I
I
I
be the incenter of
A
1
B
1
C
1
D
1
A_1B_1C_1D_1
A
1
B
1
C
1
D
1
and
A
2
A_2
A
2
the common point of line
A
1
I
A_1I
A
1
I
with plane
B
1
C
1
D
1
B_1C_1D_1
B
1
C
1
D
1
. Points
B
2
,
C
2
,
D
2
B_2, C_2, D_2
B
2
,
C
2
,
D
2
are defined similarly. Prove that
P
P
P
lies inside
A
2
B
2
C
2
D
2
A_2B_2C_2D_2
A
2
B
2
C
2
D
2
.
P23
1
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prove that 6 planes have a common point, starting with a tetrahedron
A tetrahedron
A
B
C
D
ABCD
A
BC
D
is given. The incircles of triangles
A
B
C
ABC
A
BC
and
A
B
D
ABD
A
B
D
with centers
O
1
,
O
2
O_1, O_2
O
1
,
O
2
, touch
A
B
AB
A
B
at points
T
1
,
T
2
T_1, T_2
T
1
,
T
2
. The plane
π
A
B
\pi_{AB}
π
A
B
passing through the midpoint of
T
1
T
2
T_1T_2
T
1
T
2
is perpendicular to
O
1
O
2
O_1O_2
O
1
O
2
. The planes
π
A
C
,
π
B
C
,
π
A
D
,
π
B
D
,
π
C
D
\pi_{AC},\pi_{BC}, \pi_{AD}, \pi_{BD}, \pi_{CD}
π
A
C
,
π
BC
,
π
A
D
,
π
B
D
,
π
C
D
are defined similarly. Prove that these six planes have a common point.
P22
1
Hide problems
painting the faces of an icosahedron into 5 colors ...
The faces of an icosahedron are painted into
5
5
5
colors in such a way that two faces painted into the same color have no common points, even a vertices. Prove that for any point lying inside the icosahedron the sums of the distances from this point to the red faces and the blue faces are equal.
P21
1
Hide problems
parallel chords of pairs of intersecting circles
A quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed into a circle
ω
\omega
ω
with center
O
O
O
. Let
M
1
M_1
M
1
and
M
2
M_2
M
2
be the midpoints of segments
A
B
AB
A
B
and
C
D
CD
C
D
respectively. Let
Ω
\Omega
Ω
be the circumcircle of triangle
O
M
1
M
2
OM_1M_2
O
M
1
M
2
. Let
X
1
X_1
X
1
and
X
2
X_2
X
2
be the common points of
ω
\omega
ω
and
Ω
\Omega
Ω
and
Y
1
Y_1
Y
1
and
Y
2
Y_2
Y
2
the second common points of
Ω
\Omega
Ω
with the circumcircles of triangles
C
D
M
1
CDM_1
C
D
M
1
and
A
B
M
2
ABM_2
A
B
M
2
. Prove that
X
1
X
2
/
/
Y
1
Y
2
X_1X_2 // Y_1Y_2
X
1
X
2
//
Y
1
Y
2
.
P20
1
Hide problems
locus of a circumecenter, given a circle lying inside an ellipse
Given are a circle and an ellipse lying inside it with focus
C
C
C
. Find the locus of the circumcenters of triangles
A
B
C
ABC
A
BC
, where
A
B
AB
A
B
is a chord of the circle touching the ellipse.
P19
1
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intersection point of 2 perpendiculars lies on a medial line
Let
L
L
L
and
K
K
K
be the feet of the internal and the external bisector of angle
A
A
A
of a triangle
A
B
C
ABC
A
BC
. Let
P
P
P
be the common point of the tangents to the circumcircle of the triangle at
B
B
B
and
C
C
C
. The perpendicular from
L
L
L
to
B
C
BC
BC
meets
A
P
AP
A
P
at point
Q
Q
Q
. Prove that
Q
Q
Q
lies on the medial line of triangle
L
K
P
LKP
L
K
P
.
P18
1
Hide problems
if 3 of 4 intersections are collinear then all 4 are, in a cyclic hexagon
Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a cyclic hexagon, points
K
,
L
,
M
,
N
K, L, M, N
K
,
L
,
M
,
N
be the common points of lines
A
B
AB
A
B
and
C
D
CD
C
D
,
A
C
AC
A
C
and
B
D
BD
B
D
,
A
F
AF
A
F
and
D
E
DE
D
E
,
A
E
AE
A
E
and
D
F
DF
D
F
respectively. Prove that if three of these points are collinear then the fourth point lies on the same line.
P17
1
Hide problems
hidden Simson lines
Let
O
O
O
be the circumcenter of a triangle
A
B
C
ABC
A
BC
. The projections of points
D
D
D
and
X
X
X
to the sidelines of the triangle lie on lines
ℓ
\ell
ℓ
and
L
L
L
such that
ℓ
/
/
X
O
\ell // XO
ℓ
//
XO
. Prove that the angles formed by
L
L
L
and by the diagonals of quadrilateral
A
B
C
D
ABCD
A
BC
D
are equal.
P16
1
Hide problems
construct quadrilateral, given 2 circumcenters + 2 incenters, by diagonals
The diagonals of a convex quadrilateral divide it into four triangles. Restore the quadrilateral by the circumcenters of two adjacent triangles and the incenters of two mutually opposite triangles
P15
1
Hide problems
prove that p(1 -2Rr) <=1, when sides a,b,c <=1
The sidelengths of a triangle
A
B
C
ABC
A
BC
are not greater than
1
1
1
. Prove that
p
(
1
−
2
R
r
)
p(1 -2Rr)
p
(
1
−
2
R
r
)
is not greater than
1
1
1
, where
p
p
p
is the semiperimeter,
R
R
R
and
r
r
r
are the circumradius and the inradius of
A
B
C
ABC
A
BC
.
P14
1
Hide problems
prove that 3 circles have 2 common points
Let
A
B
C
ABC
A
BC
be an acute-angled, nonisosceles triangle. Point
A
1
,
A
2
A_1, A_2
A
1
,
A
2
are symmetric to the feet of the internal and the external bisectors of angle
A
A
A
wrt the midpoint of
B
C
BC
BC
. Segment
A
1
A
2
A_1A_2
A
1
A
2
is a diameter of a circle
α
\alpha
α
. Circles
β
\beta
β
and
γ
\gamma
γ
are defined similarly. Prove that these three circles have two common points.
P13
1
Hide problems
altitudes, midpoints, concurrent lines with the medial line
Let
A
H
1
,
B
H
2
AH_1, BH_2
A
H
1
,
B
H
2
and
C
H
3
CH_3
C
H
3
be the altitudes of a triangle
A
B
C
ABC
A
BC
. Point
M
M
M
is the midpoint of
H
2
H
3
H_2H_3
H
2
H
3
. Line
A
M
AM
A
M
meets
H
2
H
1
H_2H_1
H
2
H
1
at point
K
K
K
. Prove that
K
K
K
lies on the medial line of
A
B
C
ABC
A
BC
parallel to
A
C
AC
A
C
.
P12
1
Hide problems
max No of disks so than each 2 have a common point and no 3 have it
Find the maximal number of discs which can be disposed on the plane so that each two of them have a common point and no three have it
P11
1
Hide problems
perimeter of a triangle equal to side of another triangle
Let
H
H
H
be the orthocenter of an acute-angled triangle A
B
C
BC
BC
. The perpendicular bisector to segment
B
H
BH
B
H
meets
B
A
BA
B
A
and
B
C
BC
BC
at points
A
0
,
C
0
A_0, C_0
A
0
,
C
0
respectively. Prove that the perimeter of triangle
A
0
O
C
0
A_0OC_0
A
0
O
C
0
(
O
O
O
is the circumcenter of triangle
A
B
C
ABC
A
BC
) is equal to
A
C
AC
A
C
.
P10
1
Hide problems
convex quadrilateral divided into 4 similar triangles by diagonals is tangential
The diagonals of a convex quadrilateral divide it into four similar triangles. Prove that is possible to inscribe a circle into this quadrilateral
P9
1
Hide problems
construction of 3 points on sides of an acute triangle
Let
A
B
C
ABC
A
BC
be an acute-angled triangle. Construct points
A
′
,
B
′
,
C
′
A', B', C'
A
′
,
B
′
,
C
′
on its sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
such that: -
A
′
B
′
∥
A
B
A'B' \parallel AB
A
′
B
′
∥
A
B
, -
C
′
C
C'C
C
′
C
is the bisector of angle
A
′
C
′
B
′
A'C'B'
A
′
C
′
B
′
, -
A
′
C
′
+
B
′
C
′
=
A
B
A'C' + B'C'= AB
A
′
C
′
+
B
′
C
′
=
A
B
.
P8
1
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equal angles in an isosceles trapezoid of perpendicular diagonals
Diagonals of an isosceles trapezoid
A
B
C
D
ABCD
A
BC
D
with bases
B
C
BC
BC
and
A
D
AD
A
D
are perpendicular. Let
D
E
DE
D
E
be the perpendicular from
D
D
D
to
A
B
AB
A
B
, and let
C
F
CF
CF
be the perpendicular from
C
C
C
to
D
E
DE
D
E
. Prove that angle
D
B
F
DBF
D
BF
is equal to half of angle
F
C
D
FCD
FC
D
.
P7
1
Hide problems
parallel lines, taking the symmetrics of orthocenter wrt to sides
The altitudes
A
A
1
AA_1
A
A
1
and
C
C
1
CC_1
C
C
1
of a triangle
A
B
C
ABC
A
BC
meet at point
H
H
H
. Point
H
A
H_A
H
A
is symmetric to
H
H
H
about
A
A
A
. Line
H
A
C
1
H_AC_1
H
A
C
1
meets
B
C
BC
BC
at point
C
′
C'
C
′
, point
A
′
A'
A
′
is defined similarly. Prove that
A
′
C
′
/
/
A
C
A' C' // AC
A
′
C
′
//
A
C
.
P6
1
Hide problems
lines connecting symmetric of feet of altitudes wrt other altitudes are parallel
Let
A
A
′
,
B
B
′
AA', BB'
A
A
′
,
B
B
′
and
C
C
′
CC'
C
C
′
be the altitudes of an acute-angled triangle
A
B
C
ABC
A
BC
. Points
C
a
,
C
b
C_a, C_b
C
a
,
C
b
are symmetric to
C
′
C'
C
′
wrt
A
A
′
AA'
A
A
′
and
B
B
′
BB'
B
B
′
. Points
A
b
,
A
c
,
B
c
,
B
a
A_b, A_c, B_c, B_a
A
b
,
A
c
,
B
c
,
B
a
are defined similarly. Prove that lines
A
b
B
a
,
B
c
C
b
A_bB_a, B_cC_b
A
b
B
a
,
B
c
C
b
and
C
a
A
c
C_aA_c
C
a
A
c
are parallel.
P5
1
Hide problems
double segment given, double angle wanted, around intesecting circles
Let a triangle
A
B
C
ABC
A
BC
be given. Two circles passing through
A
A
A
touch
B
C
BC
BC
at points
B
B
B
and
C
C
C
respectively. Let
D
D
D
be the second common point of these circles (
A
A
A
is closer to
B
C
BC
BC
than
D
D
D
). It is known that
B
C
=
2
B
D
BC = 2BD
BC
=
2
B
D
. Prove that
∠
D
A
B
=
2
∠
A
D
B
.
\angle DAB = 2\angle ADB.
∠
D
A
B
=
2∠
A
D
B
.
P4
1
Hide problems
startng with trisectors of angles <A, <B in a parallelogram ABCD
In a parallelogram
A
B
C
D
ABCD
A
BC
D
the trisectors of angles
A
A
A
and
B
B
B
are drawn. Let
O
O
O
be the common points of the trisectors nearest to
A
B
AB
A
B
. Let
A
O
AO
A
O
meet the second trisector of angle
B
B
B
at point
A
1
A_1
A
1
, and let
B
O
BO
BO
meet the second trisector of angle
A
A
A
at point
B
1
B_1
B
1
. Let
M
M
M
be the midpoint of
A
1
B
1
A_1B_1
A
1
B
1
. Line
M
O
MO
MO
meets
A
B
AB
A
B
at point
N
N
N
Prove that triangle
A
1
B
1
N
A_1B_1N
A
1
B
1
N
is equilateral.
P3
1
Hide problems
prove that <BGE <= < AED / 2
The side
A
D
AD
A
D
of a square
A
B
C
D
ABCD
A
BC
D
is the base of an obtuse-angled isosceles triangle
A
E
D
AED
A
E
D
with vertex
E
E
E
lying inside the square. Let
A
F
AF
A
F
be a diameter of the circumcircle of this triangle, and
G
G
G
be a point on
C
D
CD
C
D
such that
C
G
=
D
F
CG = DF
CG
=
D
F
. Prove that angle
B
G
E
BGE
BGE
is less than half of angle
A
E
D
AED
A
E
D
.
P2
1
Hide problems
easy angle chasing for 8graders
Let
O
O
O
and
H
H
H
be the circumcenter and the orthocenter of a triangle
A
B
C
ABC
A
BC
. The line passing through the midpoint of
O
H
OH
O
H
and parallel to
B
C
BC
BC
meets
A
B
AB
A
B
and
A
C
AC
A
C
at points
D
D
D
and
E
E
E
. It is known that
O
O
O
is the incenter of triangle
A
D
E
ADE
A
D
E
. Find the angles of
A
B
C
ABC
A
BC
.
P1
1
Hide problems
cutting out a convex polygonm folding it several times, can it be an 7-gon?
Tanya cut out a convex polygon from the paper, fold it several times and obtained a two-layers quadrilateral. Can the cutted polygon be a heptagon?
8
3
Hide problems
angle chasing practice for Russian 8-graders
Points
C
1
,
B
1
C_1, B_1
C
1
,
B
1
on sides
A
B
,
A
C
AB, AC
A
B
,
A
C
respectively of triangle
A
B
C
ABC
A
BC
are such that
B
B
1
⊥
C
C
1
BB_1 \perp CC_1
B
B
1
⊥
C
C
1
. Point
X
X
X
lying inside the triangle is such that
∠
X
B
C
=
∠
B
1
B
A
,
∠
X
C
B
=
∠
C
1
C
A
\angle XBC = \angle B_1BA, \angle XCB = \angle C_1CA
∠
XBC
=
∠
B
1
B
A
,
∠
XCB
=
∠
C
1
C
A
. Prove that
∠
B
1
X
C
1
=
9
0
o
−
∠
A
\angle B_1XC_1 =90^o- \angle A
∠
B
1
X
C
1
=
9
0
o
−
∠
A
.(A. Antropov, A. Yakubov)
Circle joining centers tangent to original circumcircle
A perpendicular bisector of side
B
C
BC
BC
of triangle
A
B
C
ABC
A
BC
meets lines
A
B
AB
A
B
and
A
C
AC
A
C
at points
A
B
A_B
A
B
and
A
C
A_C
A
C
respectively. Let
O
a
O_a
O
a
be the circumcenter of triangle
A
A
B
A
C
AA_BA_C
A
A
B
A
C
. Points
O
b
O_b
O
b
and
O
c
O_c
O
c
are defined similarly. Prove that the circumcircle of triangle
O
a
O
b
O
c
O_aO_bO_c
O
a
O
b
O
c
touches the circumcircle of the original triangle.
Dividing rectangle geometry
Does there exist a rectangle which can be divided into a regular hexagon with sidelength
1
1
1
and several congruent right-angled triangles with legs
1
1
1
and
3
\sqrt{3}
3
?
7
3
Hide problems
2 tangential quadrilaterals from vertices of a quadrilateral & a point on 1 side
Point
M
M
M
on side
A
B
AB
A
B
of quadrilateral
A
B
C
D
ABCD
A
BC
D
is such that quadrilaterals
A
M
C
D
AMCD
A
MC
D
and
B
M
D
C
BMDC
BM
D
C
are circumscribed around circles centered at
O
1
O_1
O
1
and
O
2
O_2
O
2
respectively. Line
O
1
O
2
O_1O_2
O
1
O
2
cuts an isosceles triangle with vertex M from angle
C
M
D
CMD
CM
D
. Prove that
A
B
C
D
ABCD
A
BC
D
is a cyclic quadrilateral.(M. Kungozhin)
segment of orthocenter - centroid, bisects segment of feet of altitudes
Let
A
B
C
ABC
A
BC
be an acute-angled, nonisosceles triangle. Altitudes
A
A
′
AA'
A
A
′
and
B
B
′
BB'
B
B
′
meet at point
H
H
H
, and the medians of triangle
A
H
B
AHB
A
H
B
meet at point
M
M
M
. Line
C
M
CM
CM
bisects segment
A
′
B
′
A'B'
A
′
B
′
. Find angle
C
C
C
.(D. Krekov)
Sphere geometry
Let
S
A
B
C
D
SABCD
S
A
BC
D
be an inscribed pyramid, and
A
A
1
AA_1
A
A
1
,
B
B
1
BB_1
B
B
1
,
C
C
1
CC_1
C
C
1
,
D
D
1
DD_1
D
D
1
be the perpendiculars from
A
A
A
,
B
B
B
,
C
C
C
,
D
D
D
to lines
S
C
SC
SC
,
S
D
SD
S
D
,
S
A
SA
S
A
,
S
B
SB
SB
respectively. Points
S
S
S
,
A
1
A_1
A
1
,
B
1
B_1
B
1
,
C
1
C_1
C
1
,
D
1
D_1
D
1
are distinct and lie on a sphere. Prove that points
A
1
A_1
A
1
,
B
1
B_1
B
1
,
C
1
C_1
C
1
and
D
1
D_1
D
1
are coplanar.
6
3
Hide problems
line perpendicular to the median of a triangle
Lines
b
b
b
and
c
c
c
passing through vertices
B
B
B
and
C
C
C
of triangle
A
B
C
ABC
A
BC
are perpendicular to sideline
B
C
BC
BC
. The perpendicular bisectors to
A
C
AC
A
C
and
A
B
AB
A
B
meet
b
b
b
and
c
c
c
at points
P
P
P
and
Q
Q
Q
respectively. Prove that line
P
Q
PQ
PQ
is perpendicular to median
A
M
AM
A
M
of triangle
A
B
C
ABC
A
BC
.(D. Prokopenko)
in a convex ABCD with perpendicular diagonals, 4 lines are concurrent
The diagonals of convex quadrilateral
A
B
C
D
ABCD
A
BC
D
are perpendicular. Points
A
′
,
B
′
,
C
′
,
D
′
A' , B' , C' , D'
A
′
,
B
′
,
C
′
,
D
′
are the circumcenters of triangles
A
B
D
,
B
C
A
,
C
D
B
,
D
A
C
ABD, BCA, CDB, DAC
A
B
D
,
BC
A
,
C
D
B
,
D
A
C
respectively. Prove that lines
A
A
′
,
B
B
′
,
C
C
′
,
D
D
′
AA' , BB' , CC' , DD'
A
A
′
,
B
B
′
,
C
C
′
,
D
D
′
concur.(A. Zaslavsky)
Circumcircle geometry
Let
H
H
H
and
O
O
O
be the orthocenter and the circumcenter of triangle
A
B
C
ABC
A
BC
. The circumcircle of triangle
A
O
H
AOH
A
O
H
meets the perpendicular bisector of
B
C
BC
BC
at point
A
1
≠
O
A_1 \neq O
A
1
=
O
. Points
B
1
B_1
B
1
and
C
1
C_1
C
1
are defined similarly. Prove that lines
A
A
1
AA_1
A
A
1
,
B
B
1
BB_1
B
B
1
and
C
C
1
CC_1
C
C
1
concur.
5
3
Hide problems
moving two triangles with 1 angle a, to construct angle a/2
Two equal hard triangles are given. One of their angles is equal to
α
\alpha
α
(these angles are marked). Dispose these triangles on the plane in such a way that the angle formed by some three vertices would be equal to
α
/
2
\alpha / 2
α
/2
. (No instruments are allowed, even a pencil.)(E. Bakayev, A. Zaslavsky)
Right-angled triangle geometry
Let
B
M
BM
BM
be a median of right-angled nonisosceles triangle
A
B
C
ABC
A
BC
(
∠
B
=
90
\angle B = 90
∠
B
=
90
), and
H
a
H_a
H
a
,
H
c
H_c
H
c
be the orthocenters of triangles
A
B
M
ABM
A
BM
,
C
B
M
CBM
CBM
respectively. Lines
A
H
c
AH_c
A
H
c
and
C
H
a
CH_a
C
H
a
meet at point
K
K
K
. Prove that
∠
M
B
K
=
90
\angle MBK = 90
∠
MB
K
=
90
.
concurrent lines with the line passing through 2 midpoints
Let
B
M
BM
BM
be a median of nonisosceles right-angled triangle
A
B
C
ABC
A
BC
(
∠
B
=
9
0
o
\angle B = 90^o
∠
B
=
9
0
o
), and
H
a
,
H
c
Ha, Hc
H
a
,
Hc
be the orthocenters of triangles
A
B
M
,
C
B
M
ABM, CBM
A
BM
,
CBM
respectively. Prove that lines
A
H
c
AH_c
A
H
c
and
C
H
a
CH_a
C
H
a
meet on the medial line of triangle
A
B
C
ABC
A
BC
.(D. Svhetsov)
4
3
Hide problems
any convex quadrilateral can be divided into 5 polygons with symmetry axes
Prove that an arbitrary convex quadrilateral can be divided into five polygons having symmetry axes.(N. Belukhov)
fixed point for Russian 9graders
A fixed triangle
A
B
C
ABC
A
BC
is given. Point
P
P
P
moves on its circumcircle so that segments
B
C
BC
BC
and
A
P
AP
A
P
intersect. Line
A
P
AP
A
P
divides triangle
B
P
C
BPC
BPC
into two triangles with incenters
I
1
I_1
I
1
and
I
2
I_2
I
2
. Line
I
1
I
2
I_1I_2
I
1
I
2
meets
B
C
BC
BC
at point
Z
Z
Z
. Prove that all lines
Z
P
ZP
ZP
pass through a fixed point.(R. Krutovsky, A. Yakubov)
Touching points geometry
Let
A
A
1
AA_1
A
A
1
,
B
B
1
BB_1
B
B
1
,
C
C
1
CC_1
C
C
1
be the altitudes of an acute-angled, nonisosceles triangle
A
B
C
ABC
A
BC
, and
A
2
A_2
A
2
,
B
2
B_2
B
2
,
C
2
C_2
C
2
be the touching points of sides
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
with the correspondent excircles. It is known that line
B
1
C
1
B_1C_1
B
1
C
1
touches the incircle of
A
B
C
ABC
A
BC
. Prove that
A
1
A_1
A
1
lies on the circumcircle of
A
2
B
2
C
2
A_2B_2C_2
A
2
B
2
C
2
.
3
3
Hide problems
isosceles 20-80-80 , M in AC, AM/MC = 1/2, H projection of C on BM, <AHB?
In triangle
A
B
C
ABC
A
BC
we have
A
B
=
B
C
,
∠
B
=
2
0
o
AB = BC, \angle B = 20^o
A
B
=
BC
,
∠
B
=
2
0
o
. Point
M
M
M
on
A
C
AC
A
C
is such that
A
M
:
M
C
=
1
:
2
AM : MC = 1 : 2
A
M
:
MC
=
1
:
2
, point
H
H
H
is the projection of
C
C
C
to
B
M
BM
BM
. Find angle
A
H
B
AHB
A
H
B
.(M. Yevdokimov)
100 discs on the plane so that each two of them have a common point ...
Let
100
100
100
discs lie on the plane in such a way that each two of them have a common point. Prove that there exists a point lying inside at least
15
15
15
of these discs.(M. Kharitonov, A. Polyansky)
Midpoint geometry
Let
A
1
A_1
A
1
,
B
1
B_1
B
1
and
C
1
C_1
C
1
be the midpoints of sides
B
C
BC
BC
,
C
A
CA
C
A
and
A
B
AB
A
B
of triangle
A
B
C
ABC
A
BC
, respectively. Points
B
2
B_2
B
2
and
C
2
C_2
C
2
are the midpoints of segments
B
A
1
BA_1
B
A
1
and
C
A
1
CA_1
C
A
1
respectively. Point
B
3
B_3
B
3
is symmetric to
C
1
C_1
C
1
wrt
B
B
B
, and
C
3
C_3
C
3
is symmetric to
B
1
B_1
B
1
wrt
C
C
C
. Prove that one of common points of circles
B
B
2
B
3
BB_2B_3
B
B
2
B
3
and
C
C
2
C
3
CC_2C_3
C
C
2
C
3
lies on the circumcircle of triangle
A
B
C
ABC
A
BC
.
2
3
Hide problems
circumcircle of A,B, and orthocenter, meets AC,BC at interior points, <C ?
A circle passing through
A
,
B
A, B
A
,
B
and the orthocenter of triangle
A
B
C
ABC
A
BC
meets sides
A
C
,
B
C
AC, BC
A
C
,
BC
at their inner points. Prove that
6
0
o
<
∠
C
<
9
0
o
60^o < \angle C < 90^o
6
0
o
<
∠
C
<
9
0
o
. (A. Blinkov)
find a point of convex ABCD so that it's projections on sides are vertices of #
A convex quadrilateral is given. Using a compass and a ruler construct a point such that its projections to the sidelines of this quadrilateral are the vertices of a parallelogram.(A. Zaslavsky)
Covering by triangle
Prove that an arbitrary triangle with area
1
1
1
can be covered by an isosceles triangle with area less than
2
\sqrt{2}
2
.
1
3
Hide problems
right trapezoid, double angle given, midpoint wanted
In trapezoid
A
B
C
D
ABCD
A
BC
D
angles
A
A
A
and
B
B
B
are right,
A
B
=
A
D
,
C
D
=
B
C
+
A
D
,
B
C
<
A
D
AB = AD, CD = BC + AD, BC < AD
A
B
=
A
D
,
C
D
=
BC
+
A
D
,
BC
<
A
D
. Prove that
∠
A
D
C
=
2
∠
A
B
E
\angle ADC = 2\angle ABE
∠
A
D
C
=
2∠
A
BE
, where
E
E
E
is the midpoint of segment
A
D
AD
A
D
.(V. Yasinsky)
intersecting circles & tangents at the common, inequality of segments wanted
Circles
α
\alpha
α
and
β
\beta
β
pass through point
C
C
C
. The tangent to
α
\alpha
α
at this point meets
β
\beta
β
at point
B
B
B
, and the tangent to
β
\beta
β
at
C
C
C
meets
α
\alpha
α
at point
A
A
A
so that
A
A
A
and
B
B
B
are distinct from
C
C
C
and angle
A
C
B
ACB
A
CB
is obtuse. Line
A
B
AB
A
B
meets
α
\alpha
α
and
β
\beta
β
for the second time at points
N
N
N
and
M
M
M
respectively. Prove that
2
M
N
<
A
B
2MN < AB
2
MN
<
A
B
.(D. Mukhin)
Geometry with bisectors
Let
K
K
K
be an arbitrary point on side
B
C
BC
BC
of triangle
A
B
C
ABC
A
BC
, and
K
N
KN
K
N
be a bisector of triangle
A
K
C
AKC
A
K
C
. Lines
B
N
BN
BN
and
A
K
AK
A
K
meet at point
F
F
F
, and lines
C
F
CF
CF
and
A
B
AB
A
B
meet at point
D
D
D
. Prove that
K
D
KD
KD
is a bisector of triangle
A
K
B
AKB
A
K
B
.