MathDB
Problems
Contests
National and Regional Contests
Russia Contests
Adygea Teachers' Geometry Olympiad
2022 Adygea Teachers' Geometry Olympiad
2022 Adygea Teachers' Geometry Olympiad
Part of
Adygea Teachers' Geometry Olympiad
Subcontests
(4)
4
1
Hide problems
plane cuts hexagonal pyramid (2022 Adygea Teachers' Geometry Olympiad p4)
In a regular hexagonal pyramid
S
A
B
C
D
E
F
SABCDEF
S
A
BC
D
EF
, a plane is drawn through vertex
A
A
A
and the midpoints of edges
S
C
SC
SC
and
C
E
CE
CE
. Find the ratio in which this plane divides the volume of the pyramid.
3
1
Hide problems
BP bisects AC, incircle (2022 Adygea Teachers' Geometry Olympiad p3)
The incircle of triangle
A
B
C
ABC
A
BC
touches its sides at points
A
′
A'
A
′
,
B
′
B'
B
′
,
C
′
C'
C
′
.
I
I
I
is its center. Straight line
B
′
I
B'I
B
′
I
intersects segment
A
′
C
′
A'C'
A
′
C
′
at point
P
P
P
. Prove that straight line
B
P
BP
BP
passes through the midpoint of
A
C
AC
A
C
.
2
1
Hide problems
fixed point, trapezoid (2022 Adygea Teachers' Geometry Olympiad p2)
An arbitrary point
P
P
P
is chosen on the lateral side
A
B
AB
A
B
of the trapezoid
A
B
C
D
ABCD
A
BC
D
. Straight lines passing through it parallel to the diagonals of the trapezoid intersect the bases at points
Q
Q
Q
and
R
R
R
. Prove that the sides
Q
R
QR
QR
of all possible triangles
P
Q
R
PQR
PQR
pass through a fixed point.
1
1
Hide problems
MN/AB =? 60-45-75 triangle (2022 Adygea Teachers' Geometry Olympiad p1)
In triangle
A
B
C
ABC
A
BC
,
∠
A
=
6
0
o
\angle A = 60^o
∠
A
=
6
0
o
,
∠
B
=
4
5
o
\angle B = 45^o
∠
B
=
4
5
o
. On the sides
A
C
AC
A
C
and
B
C
BC
BC
points
M
M
M
and
N
N
N
are taken, respectively, so that the straight line
M
N
MN
MN
cuts off a triangle similar to this one. Find the ratio of
M
N
MN
MN
to
A
B
AB
A
B
if it is known that
C
M
:
A
M
=
2
:
1
CM : AM = 2:1
CM
:
A
M
=
2
:
1
.