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Romania Team Selection Test
1997 Romania Team Selection Test
1997 Romania Team Selection Test
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Romania Team Selection Test
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Sum of f(P) where P is on circle C and in the set M is 0
Let
n
≥
4
n\ge 4
n
≥
4
be a positive integer and let
M
M
M
be a set of
n
n
n
points in the plane, where no three points are collinear and not all of the
n
n
n
points being concyclic. Find all real functions
f
:
M
→
R
f:M\to\mathbb{R}
f
:
M
→
R
such that for any circle
C
\mathcal{C}
C
containing at least three points from
M
M
M
, the following equality holds:
∑
P
∈
C
∩
M
f
(
P
)
=
0
\sum_{P\in\mathcal{C}\cap M} f(P)=0
P
∈
C
∩
M
∑
f
(
P
)
=
0
Dorel Mihet
Number of possible colourings involving dodecagon
The vertices of a regular dodecagon are coloured either blue or red. Find the number of all possible colourings which do not contain monochromatic sub-polygons.Vasile Pop
function f(x^2+y^2)=f(x^2-y^2)+f(2xy)
Find all functions
f
:
R
→
[
0
;
+
∞
)
f: \mathbb{R}\to [0;+\infty)
f
:
R
→
[
0
;
+
∞
)
such that:
f
(
x
2
+
y
2
)
=
f
(
x
2
−
y
2
)
+
f
(
2
x
y
)
f(x^2+y^2)=f(x^2-y^2)+f(2xy)
f
(
x
2
+
y
2
)
=
f
(
x
2
−
y
2
)
+
f
(
2
x
y
)
for all real numbers
x
x
x
and
y
y
y
.Laurentiu Panaitopol
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