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Novosibirsk Oral Olympiad in Geometry
2022 Novosibirsk Oral Olympiad in Geometry
2022 Novosibirsk Oral Olympiad in Geometry
Part of
Novosibirsk Oral Olympiad in Geometry
Subcontests
(7)
4
3
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<B=? if <C = 3<A and AB=2 BC (2022 Novosibirsk Oral Geo Oly 8.4)
In triangle
A
B
C
ABC
A
BC
, angle
C
C
C
is three times the angle
A
A
A
, and side
A
B
AB
A
B
is twice the side
B
C
BC
BC
. What can be the angle
A
B
C
ABC
A
BC
?
rectangle by 7 corners (2022 Novosibirsk Oral Geo Oly 7.4 8.3)
Fold the next seven corners into a rectangle. https://cdn.artofproblemsolving.com/attachments/b/b/2b8b9d6d4b72024996a66d41f865afb91bb9b7.png
incenter of BCD lies on circumcenter of ABD (2022 Novosibirsk Oral Geo Oly 9.4)
A point
D
D
D
is marked on the side
A
C
AC
A
C
of triangle
A
B
C
ABC
A
BC
. The circumscribed circle of triangle
A
B
D
ABD
A
B
D
passes through the center of the inscribed circle of triangle
B
C
D
BCD
BC
D
. Find
∠
A
C
B
\angle ACB
∠
A
CB
if
∠
A
B
C
=
4
0
o
\angle ABC = 40^o
∠
A
BC
=
4
0
o
.
5
3
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area, 2 equal rectangles (2022 Novosibirsk Oral Geo Oly 7.5)
Two equal rectangles of area
10
10
10
are arranged as follows. Find the area of the gray rectangle. https://cdn.artofproblemsolving.com/attachments/7/1/112b07530a2ef42e5b2cf83a2cb9fb11dfc9e6.png
area by 2 isosceles triangles of same area (2022 Novosibirsk Oral Geo Oly 8.5)
Two isosceles triangles of the same area are located as shown in the figure. Find the angle
x
x
x
. https://cdn.artofproblemsolving.com/attachments/a/6/f7dbfd267274781b67a5f3d5a9036fb2905156.png
triangle split into 22 triangles with 22^o (2022 Novosibirsk Oral Geo Oly 9.5)
Prove that any triangle can be divided into
22
22
22
triangles, each of which has an angle of
2
2
o
22^o
2
2
o
, and another
23
23
23
triangles, each of which has an angle of
2
3
o
23^o
2
3
o
.
7
3
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max no of several identical matches (2022 Novosibirsk Oral Geo Oly 7.7)
Vera has several identical matches, from which she makes a triangle. Vera wants any two sides of this triangle to differ in length by at least
10
10
10
matches, but it turned out that it is impossible to add such a triangle from the available matches (it is impossible to leave extra matches). What is the maximum number of matches Vera can have?
intersection of perp. diagonals lies on XY (2022 Novosibirsk Oral Geo Oly 8.7)
The diagonals of the convex quadrilateral
A
B
C
D
ABCD
A
BC
D
intersect at the point
O
O
O
. The points
X
X
X
and
Y
Y
Y
are symmetrical to the point
O
O
O
with respect to the midpoints of the sides
B
C
BC
BC
and
A
D
AD
A
D
, respectively. It is known that
A
B
=
B
C
=
C
D
AB = BC = CD
A
B
=
BC
=
C
D
. Prove that the point of intersection of the perpendicular bisectors of the diagonals of the quadrilateral lies on the line
X
Y
XY
X
Y
.
equidistant points from midpoint,orthocenter (2022 Novosibirsk Oral Geo Oly 9.7)
Altitudes
A
A
1
AA_1
A
A
1
and
C
C
1
CC_1
C
C
1
of an acute-angled triangle
A
B
C
ABC
A
BC
intersect at point
H
H
H
. A straight line passing through point
H
H
H
parallel to line
A
1
C
1
A_1C_1
A
1
C
1
intersects the circumscribed circles of triangles
A
H
C
1
AHC_1
A
H
C
1
and
C
H
A
1
CHA_1
C
H
A
1
at points
X
X
X
and
Y
Y
Y
, respectively. Prove that points
X
X
X
and
Y
Y
Y
are equidistant from the midpoint of segment
B
H
BH
B
H
.
6
3
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angle bisector wanted in 120-40-20 (2022 Novosibirsk Oral Geo Oly 7.6)
A triangle
A
B
C
ABC
A
BC
is given in which
∠
B
A
C
=
4
0
o
\angle BAC = 40^o
∠
B
A
C
=
4
0
o
. and
∠
A
B
C
=
2
0
o
\angle ABC = 20^o
∠
A
BC
=
2
0
o
. Find the length of the angle bisector drawn from the vertex
C
C
C
, if it is known that the sides
A
B
AB
A
B
and
B
C
BC
BC
differ by
4
4
4
centimeters.
cut isosceles rigth triangle in 9 triangles (2022 Novosibirsk Oral Geo Oly 8.6)
Anton has an isosceles right triangle, which he wants to cut into
9
9
9
triangular parts in the way shown in the picture. What is the largest number of the resulting
9
9
9
parts that can be equilateral triangles?A more formal description of partitioning. Let triangle
A
B
C
ABC
A
BC
be given. We choose two points on its sides so that they go in the order
A
C
1
C
2
B
A
1
A
2
C
B
1
B
2
AC_1C_2BA_1A_2CB_1B_2
A
C
1
C
2
B
A
1
A
2
C
B
1
B
2
, and no two coincide. In addition, the segments
C
1
A
2
C_1A_2
C
1
A
2
,
A
1
B
2
A_1B_2
A
1
B
2
and
B
1
C
2
B_1C_2
B
1
C
2
must intersect at one point. Then the partition is given by segments
C
1
A
2
C_1A_2
C
1
A
2
,
A
1
B
2
A_1B_2
A
1
B
2
,
B
1
C
2
B_1C_2
B
1
C
2
,
A
1
C
2
A_1C_2
A
1
C
2
,
B
1
A
2
B_1A_2
B
1
A
2
and
C
1
B
2
C_1B_2
C
1
B
2
. https://cdn.artofproblemsolving.com/attachments/0/5/5dd914b987983216342e23460954d46755d351.png
equilaterals AX/XB =BY/YC = CZ/ZA = 2/1 (2022 Novosibirsk Oral Geo Oly 9.6)
Triangle
A
B
C
ABC
A
BC
is given. On its sides
A
B
AB
A
B
,
B
C
BC
BC
and
C
A
CA
C
A
, respectively, points
X
,
Y
,
Z
X, Y, Z
X
,
Y
,
Z
are chosen so that
A
X
:
X
B
=
B
Y
:
Y
C
=
C
Z
:
Z
A
=
2
:
1.
AX : XB =BY : YC = CZ : ZA = 2:1.
A
X
:
XB
=
B
Y
:
Y
C
=
CZ
:
Z
A
=
2
:
1.
It turned out that the triangle
X
Y
Z
XYZ
X
Y
Z
is equilateral. Prove that the original triangle
A
B
C
ABC
A
BC
is also equilateral.
3
2
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angles by angle bisectors given (2022 Novosibirsk Oral Geo Oly 7.3)
Three angle bisectors were drawn in a triangle, and it turned out that the angles between them are
5
0
o
50^o
5
0
o
,
6
0
o
60^o
6
0
o
and
7
0
o
70^o
7
0
o
. Find the angles of the original triangle.
regular hexagon (2022 Novosibirsk Oral Geo Oly 9.3)
In a regular hexagon, segments with lengths from
1
1
1
to
6
6
6
were drawn as shown in the right figure (the segments go sequentially in increasing length, all the angles between them are right). Find the side length of this hexagon. https://cdn.artofproblemsolving.com/attachments/3/1/82e4225b56d984e897a43ba1f403d89e5f4736.png
2
3
Hide problems
perimeter of quad (2022 Novosibirsk Oral Geo Oly 7.2 8.1)
A quadrilateral is given, in which the lengths of some two sides are equal to
1
1
1
and
4
4
4
. Also, the diagonal of length
2
2
2
divides it into two isosceles triangles. Find the perimeter of this quadrilateral.
reflections on billiard (2022 Novosibirsk Oral Geo Oly 8.2)
A ball was launched on a rectangular billiard table at an angle of
4
5
o
45^o
4
5
o
to one of the sides. Reflected from all sides (the angle of incidence is equal to the angle of reflection), he returned to his original position . It is known that one of the sides of the table has a length of one meter. Find the length of the second side. https://cdn.artofproblemsolving.com/attachments/3/d/e0310ea910c7e3272396cd034421d1f3e88228.png
min total length by 4 integer line segments (2022 Novosibirsk Oral Geo Oly 9.2)
Faith has four different integer length segments. It turned out that any three of them can form a triangle. What is the smallest total length of this set of segments?
1
1
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cut a square with 3 lines (2022 Novosibirsk Oral Geo Oly 7.1)
Cut a square with three straight lines into three triangles and four quadrilaterals.