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National and Regional Contests
Russia Contests
Novosibirsk Oral Olympiad in Geometry
2016 Novosibirsk Oral Olympiad in Geometry
2016 Novosibirsk Oral Olympiad in Geometry
Part of
Novosibirsk Oral Olympiad in Geometry
Subcontests
(6)
5
1
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angle bisector _|_ AB, #, bisectors (2016 Novosibirsk Oral Geo Oly 8-9 p5)
In the parallelogram
C
M
N
P
CMNP
CMNP
extend the bisectors of angles
M
C
N
MCN
MCN
and
P
C
N
PCN
PCN
and intersect with extensions of sides PN and
M
N
MN
MN
at points
A
A
A
and
B
B
B
, respectively. Prove that the bisector of the original angle
C
C
C
of the the parallelogram is perpendicular to
A
B
AB
A
B
. https://cdn.artofproblemsolving.com/attachments/f/3/fde8ef133758e06b1faf8bdd815056173f9233.png
6
1
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equal distances , point in equilateral (2016 Novosibirsk Oral Geo Oly 8-9 p6)
An arbitrary point
M
M
M
inside an equilateral triangle
A
B
C
ABC
A
BC
was connected to vertices. Prove that on each side the triangle can be selected one point at a time so that the distances between them would be equal to
A
M
,
B
M
,
C
M
AM, BM, CM
A
M
,
BM
,
CM
.
4
1
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midpoint wanted, 2 squares (2016 Novosibirsk Oral Geo Oly 8-9 p4)
The two angles of the squares are adjacent, and the extension of the diagonals of one square intersect the diagonal of another square at point
O
O
O
(see figure). Prove that
O
O
O
is the midpoint of
A
B
AB
A
B
. https://cdn.artofproblemsolving.com/attachments/7/8/8daaaa55c38e15c4a8ac7492c38707f05475cc.png
3
1
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angle wanted, AQ:QC = 3:1, square grid (2016 Novosibirsk Oral Geo Oly 8-9 p3)
A square is drawn on a sheet of grid paper on the sides of the cells
A
B
C
D
ABCD
A
BC
D
with side
8
8
8
. Point
E
E
E
is the midpoint of side
B
C
BC
BC
,
Q
Q
Q
is such a point on the diagonal
A
C
AC
A
C
such that
A
Q
:
Q
C
=
3
:
1
AQ: QC = 3: 1
A
Q
:
QC
=
3
:
1
. Find the angle between straight lines
A
E
AE
A
E
and
D
Q
DQ
D
Q
.
2
1
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difference of angles wanted, equal angles(2016 Novosibirsk Oral Geo Oly 8-9 p2)
Bisector of one angle of triangle
A
B
C
ABC
A
BC
is equal to the bisector of its external angle at the same vertex (see figure). Find the difference between the other two angles of the triangle. https://cdn.artofproblemsolving.com/attachments/c/3/d2efeb65544c45a15acccab8db05c8314eb5f2.png
1
1
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AD=? ABCD, <A=<C=120^o, AB=CD=1,CB=4 (2016 Novosibirsk Oral Geo Oly 8-9 p1)
In the quadrilateral
A
B
C
D
ABCD
A
BC
D
, angles
B
B
B
and
C
C
C
are equal to
12
0
o
120^o
12
0
o
,
A
B
=
C
D
=
1
AB = CD = 1
A
B
=
C
D
=
1
,
C
B
=
4
CB = 4
CB
=
4
. Find the length
A
D
AD
A
D
.