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Yasinsky Geometry Olympiad
2022 Yasinsky Geometry Olympiad
2022 Yasinsky Geometry Olympiad
Part of
Yasinsky Geometry Olympiad
Subcontests
(6)
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cumputational geo with altitude and angle bisector, <C=1/4 <A=1/4 <B
In the triangle
A
B
C
ABC
A
BC
, angle
C
C
C
is four times smaller than each of the other two angle The altitude
A
K
AK
A
K
and the angle bisector
A
L
AL
A
L
are drawn from the vertex of the angle
A
A
A
. It is known that the length of
A
L
AL
A
L
is equal to
ℓ
\ell
ℓ
. Find the length of the segment
L
K
LK
L
K
.(Gryhoriy Filippovskyi)
perimeter of ABC wanted, BH+CH=4r
In the acute triangle
A
B
C
ABC
A
BC
, the sum of the distances from the vertices
B
B
B
and
C
C
C
to of the orthocenter
H
H
H
is equal to
4
r
,
4r,
4
r
,
where
r
r
r
is the radius of the circle inscribed in this triangle. Find the perimeter of triangle
A
B
C
ABC
A
BC
if it is known that
B
C
=
a
BC=a
BC
=
a
.(Gryhoriy Filippovskyi)
square inside a square AP/PB=BQ/QC=CR/RD=DT/TA=1/2
On the sides
A
B
AB
A
B
,
B
C
BC
BC
,
C
D
CD
C
D
,
D
A
DA
D
A
of the square
A
B
C
D
ABCD
A
BC
D
points
P
,
Q
,
R
,
T
P, Q, R, T
P
,
Q
,
R
,
T
are chosen such that
A
P
P
B
=
B
Q
Q
C
=
C
R
R
D
=
D
T
T
A
=
1
2
.
\frac{AP}{PB}=\frac{BQ}{QC}=\frac{CR}{RD}=\frac{DT}{TA}=\frac12.
PB
A
P
=
QC
BQ
=
R
D
CR
=
T
A
D
T
=
2
1
.
The segments
A
R
AR
A
R
,
B
T
BT
BT
,
C
P
CP
CP
,
D
Q
DQ
D
Q
in the intersection form the quadrilateral
K
L
M
N
KLMN
K
L
MN
(see figure). https://cdn.artofproblemsolving.com/attachments/f/c/587a2358734c300fe7082c520f90c91f872b49.pnga) Prove that
K
L
M
N
KLMN
K
L
MN
is a square. b) Find the ratio of the areas of the squares
K
L
M
N
KLMN
K
L
MN
and
A
B
C
D
ABCD
A
BC
D
.(Alexander Shkolny)
1
3
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trisect 108^o
An angle whose degree measure is equal to
10
8
o
108^o
10
8
o
is given . Describe how with help compass and ruler can divide this angle into three equal parts.(Yukhim Rabinovych)
angles wanted, AB = 2AM, AD = 4AM, median AM cuts (ABC) at D
In the triangle
A
B
C
ABC
A
BC
, the median
A
M
AM
A
M
is extended to the intersection with the circumscribed circle at point
D
D
D
. It is known that
A
B
=
2
A
M
AB = 2AM
A
B
=
2
A
M
and
A
D
=
4
A
M
AD = 4AM
A
D
=
4
A
M
. Find the angles of the triangle
A
B
C
ABC
A
BC
.(Gryhoriy Filippovskyi)
triangle of equal area construction wanted, A-excenter related
From the triangle
A
B
C
ABC
A
BC
, are gicen only the incenter
I
I
I
, the touchpoint
K
K
K
of the inscribed circle with the side
A
B
AB
A
B
, as well as the center
I
a
I_a
I
a
of the exscribed circle, that touches the side
B
C
BC
BC
. Construct a triangle equal in size to triangle
A
B
C
ABC
A
BC
.(Gryhoriy Filippovskyi)