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Novosibirsk Oral Olympiad in Geometry
2020 Novosibirsk Oral Olympiad in Geometry
2020 Novosibirsk Oral Olympiad in Geometry
Part of
Novosibirsk Oral Olympiad in Geometry
Subcontests
(7)
7
3
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min n, n equal triangles in convex n-gon (2020 Novosibirsk Oral Geo Oly 7.7)
The segments connecting the interior point of a convex non-sided
n
n
n
-gon with its vertices divide the
n
n
n
-gon into
n
n
n
congruent triangles. For what is the smallest
n
n
n
that is possible?
bisector wanted, 30^o, 150^o, AB=BD (2020 Novosibirsk Oral Geo Oly 8.7)
You are given a quadrilateral
A
B
C
D
ABCD
A
BC
D
. It is known that
∠
B
A
C
=
3
0
o
\angle BAC = 30^o
∠
B
A
C
=
3
0
o
,
∠
D
=
15
0
o
\angle D = 150^o
∠
D
=
15
0
o
and, in addition,
A
B
=
B
D
AB = BD
A
B
=
B
D
. Prove that
A
C
AC
A
C
is the bisector of angle
C
C
C
.
AD=AB+CD, tangent circle to 3 sides, cyclic (2020 Novosibirsk Oral Geo Oly 9.7)
The quadrilateral
A
B
C
D
ABCD
A
BC
D
is known to be inscribed in a circle, and that there is a circle with center on side
A
D
AD
A
D
tangent to the other three sides. Prove that
A
D
=
A
B
+
C
D
AD = AB + CD
A
D
=
A
B
+
C
D
.
6
2
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<A'B'C'=? angle bisectors, <B=120^o (2020 Novosibirsk Oral Geo Oly 7.6)
Angle bisectors
A
A
′
,
B
B
′
AA', BB'
A
A
′
,
B
B
′
and
C
C
′
CC'
C
C
′
are drawn in triangle
A
B
C
ABC
A
BC
with angle
∠
B
=
12
0
o
\angle B= 120^o
∠
B
=
12
0
o
. Find
∠
A
′
B
′
C
′
\angle A'B'C'
∠
A
′
B
′
C
′
.
equal angles, tangents of circumcircle (2020 Novosibirsk Oral Geo Oly 9.6)
In triangle
A
B
C
ABC
A
BC
, point
M
M
M
is the midpoint of
B
C
BC
BC
,
P
P
P
the point of intersection of the tangents at points
B
B
B
and
C
C
C
of the circumscribed circle of
A
B
C
ABC
A
BC
,
N
N
N
is the midpoint of the segment
M
P
MP
MP
. The segment
A
N
AN
A
N
meets the circumcircle
A
B
C
ABC
A
BC
at the point
Q
Q
Q
. Prove that
∠
P
M
Q
=
∠
M
A
Q
\angle PMQ = \angle MAQ
∠
PMQ
=
∠
M
A
Q
.
5
2
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CK_|_AB if AK=KB+PC, BC=AP, <APC +<ABC=180^o (2020 Novosibirsk Oral Geo Oly 7.5)
Point
P
P
P
is chosen inside triangle
A
B
C
ABC
A
BC
so that
∠
A
P
C
+
∠
A
B
C
=
18
0
o
\angle APC+\angle ABC=180^o
∠
A
PC
+
∠
A
BC
=
18
0
o
and
B
C
=
A
P
.
BC=AP.
BC
=
A
P
.
On the side
A
B
AB
A
B
, a point
K
K
K
is chosen such that
A
K
=
K
B
+
P
C
AK = KB + PC
A
K
=
K
B
+
PC
. Prove that
C
K
⊥
A
B
CK \perp AB
C
K
⊥
A
B
.
midpoint wanted, perp. bisectors (2020 Novosibirsk Oral Geo Oly 8.5)
Line
ℓ
\ell
ℓ
is perpendicular to one of the medians of the triangle. The median perpendiculars to the sides of this triangle intersect the line
ℓ
\ell
ℓ
at three points. Prove that one of them is the midpoint of the segment formed by the other two.
4
2
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perp. bisector of NM bisects AB, altitudes (2020 Novosibirsk Oral Geo Oly 7.4)
The altitudes
A
N
AN
A
N
and
B
M
BM
BM
are drawn in triangle
A
B
C
ABC
A
BC
. Prove that the perpendicular bisector to the segment
N
M
NM
NM
divides the segment
A
B
AB
A
B
in half.
equal angles in square, midpoints (2020 Novosibirsk Oral Geo Oly 9.4)
Points
E
E
E
and
F
F
F
are the midpoints of sides
B
C
BC
BC
and
C
D
CD
C
D
of square
A
B
C
D
ABCD
A
BC
D
, respectively. Lines
A
E
AE
A
E
and
B
F
BF
BF
meet at point
P
P
P
. Prove that
∠
P
D
A
=
∠
A
E
D
\angle PDA = \angle AED
∠
P
D
A
=
∠
A
E
D
.
3
2
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cut an triangle into 2019 pieces (2020 Novosibirsk Oral Geo Oly 7.3)
Cut an arbitrary triangle into
2019
2019
2019
pieces so that one of them turns out to be a triangle, one is a quadrilateral, ... one is a
2019
2019
2019
-gon and one is a
2020
2020
2020
-gon. Polygons do not have to be convex.
3 statements, one false, right triangle (2020 Novosibirsk Oral Geo Oly 8.3)
Maria Ivanovna drew on the blackboard a right triangle
A
B
C
ABC
A
BC
with a right angle
B
B
B
. Three students looked at her and said:
∙
\bullet
∙
Yura said: "The hypotenuse of this triangle is
10
10
10
cm."
∙
\bullet
∙
Roma said: "The altitude drawn from the vertex
B
B
B
on the side
A
C
AC
A
C
is
6
6
6
cm."
∙
\bullet
∙
Seva said: "The area of the triangle
A
B
C
ABC
A
BC
is
25
25
25
cm
2
^2
2
." Determine which of the students was mistaken if it is known that there is exactly one such person.
2
3
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max no of pieces, same perimeter, chessboard (2020 Novosibirsk Oral Geo Oly 8.2)
Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get?
3 /4 sides of any quadr., create triangle? (2020 Novosibirsk Oral Geo Oly 7.2)
It is known that four of these sticks can be assembled into a quadrilateral. Is it always true that you can make a triangle out of three of them?
cut diagonal of 2x2 in 6 equal parts, ruler (2020 Novosibirsk Oral Geo Oly 9.1)
A
2
×
2
2 \times 2
2
×
2
square was cut out of a sheet of grid paper. Using only a ruler without divisions and without going beyond the square, divide the diagonal of the square into
6
6
6
equal parts.
1
3
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areas by lines in regular 12-gon (2020 Novosibirsk Oral Geo Oly 7.1)
All twelve points on the circle are at equal distances. The only marked point inside is the center of the circle. Determine which part of the whole circle in the picture is filled in. https://cdn.artofproblemsolving.com/attachments/9/0/9a6af9cef6a4bb03fb4d3eef715f3fd77c74b3.png
3 squares inscribed in a triangle (2020 Novosibirsk Oral Geo Oly 8.1)
Three squares of area
4
,
9
4, 9
4
,
9
and
36
36
36
are inscribed in the triangle as shown in the figure. Find the area of the big triangle https://cdn.artofproblemsolving.com/attachments/9/7/3e904a9c78307e1be169ec0b95b1d3d24c1aa2.png
tangent semicircles inside a rectangle (2020 Novosibirsk Oral Geo Oly 9.1)
Two semicircles touch the side of the rectangle, each other and the segment drawn in it as in the figure. What part of the whole rectangle is filled? https://cdn.artofproblemsolving.com/attachments/3/e/70ca8b80240a282553294a58cb3ed807d016be.png