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Contests
National and Regional Contests
Lithuania Contests
Grand Duchy of Lithuania
2020 Grand Duchy of Lithuania
2020 Grand Duchy of Lithuania
Part of
Grand Duchy of Lithuania
Subcontests
(4)
2
1
Hide problems
road around 100 cities in Matland
There are
100
100
100
cities in Matland. Every road in Matland connects two cities, does not pass through any other city and does not form crossroads with other roads (although roads can go through tunnels one after the other). Driving in Matlandia by road, it is possible to get from any city to any other. Prove that that it is possible to repair some of the roads of Matlandia so that from an odd number of repaired roads would go in each city.
4
1
Hide problems
n = a^2 + b^3 + c^3 + d^5
We shall call an integer n cute if it can be written in the form
n
=
a
2
+
b
3
+
c
3
+
d
5
n = a^2 + b^3 + c^3 + d^5
n
=
a
2
+
b
3
+
c
3
+
d
5
, where
a
,
b
,
c
a, b, c
a
,
b
,
c
and
d
d
d
are integers. a) Determine if the number
2020
2020
2020
is cute. b) Find all cute integers
1
1
Hide problems
f (xf (y) - yf (x)) = f (xy) - xy
Find all functions
f
:
R
→
R
f: R \to R
f
:
R
→
R
, such that equality
f
(
x
f
(
y
)
−
y
f
(
x
)
)
=
f
(
x
y
)
−
x
y
f (xf (y) - yf (x)) = f (xy) - xy
f
(
x
f
(
y
)
−
y
f
(
x
))
=
f
(
x
y
)
−
x
y
holds for all
x
,
y
∈
R
x, y \in R
x
,
y
∈
R
.
3
1
Hide problems
altitudes of ADE intersect at midpoint of BC
The tangents of the circumcircle
Ω
\Omega
Ω
of the triangle
A
B
C
ABC
A
BC
at points
B
B
B
and
C
C
C
intersect at point
P
P
P
. The perpendiculars drawn from point
P
P
P
to lines
A
B
AB
A
B
and
A
C
AC
A
C
intersect at points
D
D
D
and
E
E
E
respectively. Prove that the altitudes of the triangle
A
D
E
ADE
A
D
E
intersect at the midpoint of the segment
B
C
BC
BC
.