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Contests
National and Regional Contests
Lithuania Contests
Grand Duchy of Lithuania
2023 Grand Duchy of Lithuania
2023 Grand Duchy of Lithuania
Part of
Grand Duchy of Lithuania
Subcontests
(4)
4
1
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(2^n -1)!! -1 divides 2^n
Note that
k
≥
1
k\ge 1
k
≥
1
for an odd natural number
k
!
!
=
k
⋅
(
k
−
2
)
⋅
.
.
.
⋅
1.
k! ! = k \cdot (k - 2) \cdot ... \cdot 1.
k
!!
=
k
⋅
(
k
−
2
)
⋅
...
⋅
1.
Prove that
(
2
n
−
1
)
!
!
−
1
(2^n -1)!! -1
(
2
n
−
1
)!!
−
1
divides
2
n
2^n
2
n
for all
n
≥
3
n \ge 3
n
≥
3
.
3
1
Hide problems
<BAK = <CAL wanted, 3 midpoints, centroid, 2 circumcircles,
The midpoints of the sides
B
C
BC
BC
,
C
A
CA
C
A
and
A
B
AB
A
B
of triangle
A
B
C
ABC
A
BC
are
M
M
M
,
N
N
N
and
P
P
P
respectively .
G
G
G
is the intersection point of the medians. The circumscribed circle around
B
G
P
BGP
BGP
intersects the line
M
P
MP
MP
at the point
K
K
K
(different than
P
P
P
).The circle circumscribed around
C
G
N
CGN
CGN
intersects the line
M
N
MN
MN
at point
L
L
L
(different than
N
N
N
). Prove that
∠
B
A
K
=
∠
C
A
L
\angle BAK = \angle CAL
∠
B
A
K
=
∠
C
A
L
.
1
1
Hide problems
f(f(x + y)) = f(x + y) + f(x)f(y) + axy
Given a non-zero real number
a
a
a
. Find all functions
f
:
R
→
R
f : R \to R
f
:
R
→
R
, such that
f
(
f
(
x
+
y
)
)
=
f
(
x
+
y
)
+
f
(
x
)
f
(
y
)
+
a
x
y
f(f(x + y)) = f(x + y) + f(x)f(y) + axy
f
(
f
(
x
+
y
))
=
f
(
x
+
y
)
+
f
(
x
)
f
(
y
)
+
a
x
y
for all
x
,
y
∈
R
x, y \in R
x
,
y
∈
R
.
2
1
Hide problems
Extremal graph
There are
n
n
n
students in a class, and some pairs of these students are friends. Among any six students, there are two of them that are not friends, and for any pair of students that are not friends there is a student among the remaining four that is friends with both of them. Find the maximum value of
n
n
n
.