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National and Regional Contests
Lithuania Contests
Grand Duchy of Lithuania
2014 Grand Duchy of Lithuania
2014 Grand Duchy of Lithuania
Part of
Grand Duchy of Lithuania
Subcontests
(4)
3
1
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bw unit squares in nxn table
In a table
n
×
n
n\times n
n
×
n
some unit squares are coloured black and the other unit squares are coloured white. For each pair of columns and each pair of rows the four squares on the intersections of these rows and columns must not all be of the same colour. What is the largest possible value of
n
n
n
?
1
1
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f(xy + f(x)) = xf(y) + f(x)
Determine all functions
f
:
R
→
R
f : R \to R
f
:
R
→
R
such that
f
(
x
y
+
f
(
x
)
)
=
x
f
(
y
)
+
f
(
x
)
f(xy + f(x)) = xf(y) + f(x)
f
(
x
y
+
f
(
x
))
=
x
f
(
y
)
+
f
(
x
)
holds for any
x
,
y
∈
R
x, y \in R
x
,
y
∈
R
.
2
1
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Prove that APM + BPC = 180
An isosceles triangle
A
B
C
ABC
A
BC
with
A
C
=
B
C
AC = BC
A
C
=
BC
is given. Let
M
M
M
be the midpoint of the side
A
B
AB
A
B
and let
P
P
P
be a point inside the triangle such that
∠
P
A
B
=
∠
P
B
C
\angle PAB = \angle PBC
∠
P
A
B
=
∠
PBC
. Prove that \angle APM + \angle BPC = 180 \textdegree
4
1
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Easy problem
Determine all positive integers
n
>
1
n > 1
n
>
1
for which
n
+
D
(
n
)
n + D(n)
n
+
D
(
n
)
is a power of
10
10
10
, where
D
(
n
)
D(n)
D
(
n
)
denotes the largest integer divisor of
n
n
n
satisfying
D
(
n
)
<
n
D(n) < n
D
(
n
)
<
n
.