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Novosibirsk Oral Olympiad in Geometry
2021 Novosibirsk Oral Olympiad in Geometry
2021 Novosibirsk Oral Olympiad in Geometry
Part of
Novosibirsk Oral Olympiad in Geometry
Subcontests
(7)
7
2
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area by 2 congrurent rectangles (2021 Novosibirsk Oral Geo Oly 7.7)
Two congruent rectangles are located as shown in the figure. Find the area of the shaded part. https://cdn.artofproblemsolving.com/attachments/2/e/10b164535ab5b3a3b98ce1a0b84892cd11d76f.png
A_1B_1//<B bisector, circle with center I (2021 Novosibirsk Oral Geo Oly 9.7)
A circle concentric with the inscribed circle of
A
B
C
ABC
A
BC
intersects the sides of the triangle at six points forming a convex hexagon
A
1
A
2
B
1
B
2
C
1
C
2
A_1A_2B_1B_2C_1C_2
A
1
A
2
B
1
B
2
C
1
C
2
(points
C
1
C_1
C
1
and
C
2
C_2
C
2
on the
A
B
AB
A
B
side,
A
1
A_1
A
1
and
A
2
A_2
A
2
on
B
C
BC
BC
,
B
1
B_1
B
1
and
B
2
B_2
B
2
on
A
C
AC
A
C
). Prove that if line
A
1
B
1
A_1B_1
A
1
B
1
is parallel to the bisector of angle
B
B
B
, then line
A
2
C
2
A_2C_2
A
2
C
2
is parallel to the bisector of angle
C
C
C
.
6
1
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AQ=BP if AP=PQ=QC, <PBQ = 30^o, equilateral (2021 Novosibirsk Oral Geo Oly 7.6)
Inside the equilateral triangle
A
B
C
ABC
A
BC
, points
P
P
P
and
Q
Q
Q
are chosen so that the quadrilateral
A
P
Q
C
APQC
A
PQC
is convex,
A
P
=
P
Q
=
Q
C
AP = PQ = QC
A
P
=
PQ
=
QC
and
∠
P
B
Q
=
3
0
o
\angle PBQ = 30^o
∠
PBQ
=
3
0
o
. Prove that
A
Q
=
B
P
AQ = BP
A
Q
=
BP
.
5
3
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<BCA=? if 2AP=BC, BX=BY, symmetrics (2021 Novosibirsk Oral Geo Oly 7.5)
In an acute-angled triangle
A
B
C
ABC
A
BC
on the side
A
C
AC
A
C
, point
P
P
P
is chosen in such a way that
2
A
P
=
B
C
2AP = BC
2
A
P
=
BC
. Points
X
X
X
and
Y
Y
Y
are symmetric to
P
P
P
with respect to vertices
A
A
A
and
C
C
C
, respectively. It turned out that
B
X
=
B
Y
BX = BY
BX
=
B
Y
. Find
∠
B
C
A
\angle BCA
∠
BC
A
.
BP=PQ wanted, CD=CE, right isosceles (2021 Novosibirsk Oral Geo Oly 8.5)
On the legs
A
C
AC
A
C
and
B
C
BC
BC
of an isosceles right-angled triangle with a right angle
C
C
C
, points
D
D
D
and
E
E
E
are taken, respectively, so that
C
D
=
C
E
CD = CE
C
D
=
CE
. Perpendiculars on line
A
E
AE
A
E
from points
C
C
C
and
D
D
D
intersect segment
A
B
AB
A
B
at points
P
P
P
and
Q
Q
Q
, respectively. Prove that
B
P
=
P
Q
BP = PQ
BP
=
PQ
.
<CNK=? <ECD=40^o, cyclic ABCDE (2021 Novosibirsk Oral Geo Oly 9.5)
The pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
is inscribed in the circle. Line segments
A
C
AC
A
C
and
B
D
BD
B
D
intersect at point
K
K
K
. Line segment
C
E
CE
CE
touches the circumcircle of triangle
A
B
K
ABK
A
B
K
at point
N
N
N
. Find the angle
C
N
K
CNK
CN
K
if
∠
E
C
D
=
4
0
o
.
\angle ECD = 40^o.
∠
EC
D
=
4
0
o
.
4
3
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are these triangles always congruent ? (2021 Novosibirsk Oral Geo Oly 7.4)
It is known about two triangles that for each of them the sum of the lengths of any two of its sides is equal to the sum of the lengths of any two sides of the other triangle. Are triangles necessarily congruent?
angle wanted, angle bisectors given (2021 Novosibirsk Oral Geo Oly 8.4)
Angle bisectors
A
D
AD
A
D
and
B
E
BE
BE
are drawn in triangle
A
B
C
ABC
A
BC
. It turned out that
D
E
DE
D
E
is the bisector of triangle
A
D
C
ADC
A
D
C
. Find the angle
B
A
C
BAC
B
A
C
.
tangent semicircle and 1/4 circle in square (2021 Novosibirsk Oral Geo Oly 9.4)
A semicircle of radius
5
5
5
and a quarter of a circle of radius
8
8
8
touch each other and are located inside the square as shown in the figure. Find the length of the part of the common tangent, enclosed in the same square. https://cdn.artofproblemsolving.com/attachments/f/2/010f501a7bc1d34561f2fe585773816f168e93.png
3
3
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angle chasing in a figure (2021 Novosibirsk Oral Geo Oly 8.3)
Find the angle
B
C
A
BCA
BC
A
in the quadrilateral of the figure. https://cdn.artofproblemsolving.com/attachments/0/2/974e23be54125cde8610a78254b59685833b5b.png
criterion for sides a=2b in a triangle (2021 Novosibirsk Oral Geo Oly 7.3)
Prove that in a triangle one of the sides is twice as large as the other if and only if a median and an angle bisector of this triangle are perpendicular
perimeter wanted, bisectors _|_ medians (2021 Novosibirsk Oral Geo Oly 9.3)
In triangle
A
B
C
ABC
A
BC
, side
A
B
AB
A
B
is
1
1
1
. It is known that one of the angle bisectors of triangle
A
B
C
ABC
A
BC
is perpendicular to one of its medians, and some other angle bisector is perpendicular to the other median. What can be the perimeter of triangle
A
B
C
ABC
A
BC
?
2
2
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angle chasing in ABCD, 80^o, 20^o, 20^o (2021 Novosibirsk Oral Geo Oly 7.2)
The extensions of two opposite sides of the convex quadrilateral intersect and form an angle of
2
0
o
20^o
2
0
o
, the extensions of the other two sides also intersect and form an angle of
2
0
o
20^o
2
0
o
. It is known that exactly one angle of the quadrilateral is
8
0
o
80^o
8
0
o
. Find all of its other angles.
flags by a robot concyclic, equal angle (2021 Novosibirsk Oral Geo Oly 9.2)
The robot crawls the meter in a straight line, puts a flag on and turns by an angle
a
<
18
0
o
a <180^o
a
<
18
0
o
clockwise. After that, everything is repeated. Prove that all flags are on the same circle.
1
2
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cut rectangle into squares, only 2 are odd (2021 Novosibirsk Oral Geo Oly 7.1)
Cut the
9
×
10
9 \times 10
9
×
10
grid rectangle along the grid lines into several squares so that there are exactly two of them with odd sidelengths.
cut rectangle into squares, only 4 are odd (2021 Novosibirsk Oral Geo Oly 9.1)
Cut the
19
×
20
19 \times 20
19
×
20
grid rectangle along the grid lines into several squares so that there are exactly four of them with odd sidelengths.