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National and Regional Contests
Russia Contests
Adygea Teachers' Geometry Olympiad
2017 Adygea Teachers' Geometry Olympiad
2017 Adygea Teachers' Geometry Olympiad
Part of
Adygea Teachers' Geometry Olympiad
Subcontests
(4)
4
1
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volume of pentahedron (2017 Adygea Teachers' Geometry Olympiad p4 RU)
A regular tetrahedron
S
A
B
C
SABC
S
A
BC
of volume
V
V
V
is given. The midpoints
D
D
D
and
E
E
E
are taken on
S
A
SA
S
A
and
S
B
SB
SB
respectively and the point
F
F
F
is taken on the edge
S
C
SC
SC
such that
S
F
:
F
C
=
1
:
3
SF: FC = 1: 3
SF
:
FC
=
1
:
3
. Find the volume of the pentahedron
F
D
E
A
B
C
FDEABC
F
D
E
A
BC
.
3
1
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3 of 4 sticks are sidelengths (2017 Adygea Teachers' Geometry Olympiad p3 RU)
Jack has a quadrilateral that consists of four sticks. It turned out that Jack can form three different triangles from those sticks. Prove that he can form a fourth triangle that is different from the others.
2
1
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exradii problem (2017 Adygea Teachers' Geometry Olympiad p2 RU)
It turned out for some triangle with sides
a
,
b
a, b
a
,
b
and
c
c
c
, that a circle of radius
r
=
a
+
b
+
c
2
r = \frac{a+b+c}{2}
r
=
2
a
+
b
+
c
touches side
c
c
c
and extensions of sides
a
a
a
and
b
b
b
. Prove that a circle of radius
a
+
c
−
b
2
\frac{a+c-b}{2}
2
a
+
c
−
b
is tangent to
a
a
a
and the extensions of
b
b
b
and
c
c
c
.
1
1
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area of trapezoid (2017 Adygea Teachers' Geometry Olympiad p1 RU)
Find the area of the
M
N
R
K
MNRK
MNR
K
trapezoid with the lateral side
R
K
=
3
RK = 3
R
K
=
3
if the distances from the vertices
M
M
M
and
N
N
N
to the line
R
K
RK
R
K
are
5
5
5
and
7
7
7
, respectively.