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Problems
Contests
National and Regional Contests
Lithuania Contests
Grand Duchy of Lithuania
2012 Grand Duchy of Lithuania
2012 Grand Duchy of Lithuania
Part of
Grand Duchy of Lithuania
Subcontests
(4)
4
1
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a_n + a_{n+1} + a_{n+m }= 0 for all bounded sequences of integers a_n
Let
m
m
m
be a positive integer. Find all bounded sequences of integers
a
1
,
a
2
,
a
3
,
.
.
.
a_1, a_2, a_3,...
a
1
,
a
2
,
a
3
,
...
for which
a
n
+
a
n
+
1
+
a
n
+
m
=
0
a_n + a_{n+1} + a_{n+m }= 0
a
n
+
a
n
+
1
+
a
n
+
m
=
0
for all
n
∈
N
n \in N
n
∈
N
.
3
1
Hide problems
line up 19 girls (all of different heights) in a row
How many ways are there to line up
19
19
19
girls (all of different heights) in a row so that no girl has a shorter girl both in front of and behind her?
1
1
Hide problems
f(x + y) = f(x)g(y) + f(y), strictly increasing f
Find all functions
g
:
R
→
R
g : R \to R
g
:
R
→
R
, for which there exists a strictly increasing function
f
:
R
→
R
f : R \to R
f
:
R
→
R
such that
f
(
x
+
y
)
=
f
(
x
)
g
(
y
)
+
f
(
y
)
f(x + y) = f(x)g(y) + f(y)
f
(
x
+
y
)
=
f
(
x
)
g
(
y
)
+
f
(
y
)
.
2
1
Hide problems
AE/BE =AC xCD / (AC^2 - CD^2) , right trapezoid with perp. diagonals
The base
A
B
AB
A
B
of a trapezium
A
B
C
D
ABCD
A
BC
D
is longer than the base
C
D
CD
C
D
, and
∠
A
D
C
\angle ADC
∠
A
D
C
is a right angle. The diagonals
A
C
AC
A
C
and
B
D
BD
B
D
are perpendicular. Let
E
E
E
be the foot of the altitude from
D
D
D
to the line
B
C
BC
BC
. Prove that
A
E
B
E
=
A
C
⋅
C
D
A
C
2
−
C
D
2
\frac{AE}{BE} =\frac{ AC \cdot CD}{AC^2 - CD^2}
BE
A
E
=
A
C
2
−
C
D
2
A
C
⋅
C
D
.