MathDB
Problems
Contests
National and Regional Contests
Ukraine Contests
Random Geometry Problems from Ukrainian Contests
Ukrainian Geometry Olympiad
2020 Ukrainian Geometry Olympiad - December
2020 Ukrainian Geometry Olympiad - December
Part of
Ukrainian Geometry Olympiad
Subcontests
(5)
4
1
Hide problems
computational inside a 80-80-20 triangle, BC=12, ABE= 30^o , EF = FC
In an isosceles triangle
A
B
C
ABC
A
BC
with an angle
∠
A
=
2
0
o
\angle A= 20^o
∠
A
=
2
0
o
and base
B
C
=
12
BC=12
BC
=
12
point
E
E
E
on the side
A
C
AC
A
C
is chosen such that
∠
A
B
E
=
3
0
o
\angle ABE= 30^o
∠
A
BE
=
3
0
o
, and point
F
F
F
on the side
A
B
AB
A
B
such that
E
F
=
F
C
EF = FC
EF
=
FC
. Find the length of
F
C
FC
FC
.
1
1
Hide problems
angle chasing in a quadrialteral with 3 equal sides, 90^o, 150^o
The three sides of the quadrilateral are equal, the angles between them are equal, respectively
9
0
o
90^o
9
0
o
and
15
0
o
150^o
15
0
o
. Find the smallest angle of this quadrilateral in degrees.
5
4
Show problems
3
4
Show problems
2
3
Hide problems
half of triangles with vertices n points on a circle are acute
On a circle noted
n
n
n
points. It turned out that among the triangles with vertices in these points exactly half of the acute. Find all values
n
n
n
in which this is possible.
max numbers of isosceles triangles by 100 collinear point and 1 outside
On a straight line lie
100
100
100
points and another point outside the line. Which is the biggest the number of isosceles triangles can be formed from the vertices of these
101
101
101
points?
S (ABC): S (ADC) wanted, cyclic BACD, AC =56, BD = 65, BC>DA, AB: BC =CD:DA
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral such that
A
C
=
56
,
B
D
=
65
,
B
C
>
D
A
AC =56, BD = 65, BC>DA
A
C
=
56
,
B
D
=
65
,
BC
>
D
A
and
A
B
:
B
C
=
C
D
:
D
A
AB: BC =CD: DA
A
B
:
BC
=
C
D
:
D
A
. Find the ratio of areas
S
(
A
B
C
)
:
S
(
A
D
C
)
S (ABC): S (ADC)
S
(
A
BC
)
:
S
(
A
D
C
)
.