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Novosibirsk Oral Olympiad in Geometry
2017 Novosibirsk Oral Olympiad in Geometry
2017 Novosibirsk Oral Olympiad in Geometry
Part of
Novosibirsk Oral Olympiad in Geometry
Subcontests
(7)
1
1
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Petya goes to Vasya using shortest route (2017 Novosibirsk Oral Geo Oly 8-9 p1)
Petya and Vasya live in neighboring houses (see the plan in the figure). Vasya lives in the fourth entrance. It is known that Petya runs to Vasya by the shortest route (it is not necessary walking along the sides of the cells) and it does not matter from which side he runs around his house. Determine in which entrance he lives Petya . https://cdn.artofproblemsolving.com/attachments/b/1/741120341a54527b179e95680aaf1c4b98ff84.png
4
1
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sum of 2 medians = semiperimeter, grid (2017 Novosibirsk Oral Geo Oly 8-9 p4)
On grid paper, mark three nodes so that in the triangle they formed, the sum of the two smallest medians equals to half-perimeter.
7
1
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sum of angles <1100^o, driver + fence (2017 Novosibirsk Oral Geo Oly 8-9 p7)
A car is driving along a straight highway at a speed of
60
60
60
km per hour. Not far from the highway there is a parallel to him a
100
100
100
-meter fence. Every second, the passenger of the car measures the angle at which the fence is visible. Prove that the sum of all the angles he measured is less than
110
0
o
1100^o
110
0
o
6
1
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ratio of bases in trapezoid ABCD, DC=DK (2017 Novosibirsk Oral Geo Oly 8-9 p6)
In trapezoid
A
B
C
D
ABCD
A
BC
D
, diagonal
A
C
AC
A
C
is the bisector of angle
A
A
A
. Point
K
K
K
is the midpoint of diagonal
A
C
AC
A
C
. It is known that
D
C
=
D
K
DC = DK
D
C
=
DK
. Find the ratio of the bases
A
D
:
B
C
AD: BC
A
D
:
BC
.
5
1
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AK+BM=CM in rectangle, CK=BC (2017 Novosibirsk Oral Geo Oly 8-9 p5)
Point
K
K
K
is marked on the diagonal
A
C
AC
A
C
in rectangle
A
B
C
D
ABCD
A
BC
D
so that
C
K
=
B
C
CK = BC
C
K
=
BC
. On the side
B
C
BC
BC
, point
M
M
M
is marked so that
K
M
=
C
M
KM = CM
K
M
=
CM
. Prove that
A
K
+
B
M
=
C
M
AK + BM = CM
A
K
+
BM
=
CM
.
3
1
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broken line's length = perimeter of ABC (2017 Novosibirsk Oral Geo Oly 8-9 p3)
Medians
A
A
1
,
B
B
1
,
C
C
1
AA_1, BB_1, CC_1
A
A
1
,
B
B
1
,
C
C
1
and altitudes
A
A
2
,
B
B
2
,
C
C
2
AA_2, BB_2, CC_2
A
A
2
,
B
B
2
,
C
C
2
are drawn in triangle
A
B
C
ABC
A
BC
. Prove that the length of the broken line
A
1
B
2
C
1
A
2
B
1
C
2
A
1
A_1B_2C_1A_2B_1C_2A_1
A
1
B
2
C
1
A
2
B
1
C
2
A
1
is equal to the perimeter of triangle
A
B
C
ABC
A
BC
.
2
1
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<CDB? <CAD=<DBA=40,<CAB=60,<CBD=20 (2017 Novosibirsk Oral Geo Oly 8-9 p2)
You are given a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
. It is known that
∠
C
A
D
=
∠
D
B
A
=
4
0
o
\angle CAD = \angle DBA = 40^o
∠
C
A
D
=
∠
D
B
A
=
4
0
o
,
∠
C
A
B
=
6
0
o
\angle CAB = 60^o
∠
C
A
B
=
6
0
o
,
∠
C
B
D
=
2
0
o
\angle CBD = 20^o
∠
CB
D
=
2
0
o
. Find the angle
∠
C
D
B
\angle CDB
∠
C
D
B
.