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Contests
National and Regional Contests
Bulgaria Contests
Bulgaria Team Selection Test
2003 Bulgaria Team Selection Test
2003 Bulgaria Team Selection Test
Part of
Bulgaria Team Selection Test
Subcontests
(6)
3
1
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a convex n gon
Some of the vertices of a convex
n
n
n
-gon are connected by segments, such that any two of them have no common interior point. Prove that, for any
n
n
n
points in general position, there exists a one-to-one correspondence between the points and the vertices of the
n
n
n
gon, such that any two segments between the points, corresponding to the respective segments from the
n
n
n
gon, have no common interior point.
5
1
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Prove a angular equality
Let
A
B
C
D
ABCD
A
BC
D
be a circumscribed quadrilateral and let
P
P
P
be the orthogonal projection of its in center on
A
C
AC
A
C
. Prove that
∠
A
P
B
=
∠
A
P
D
\angle {APB}=\angle {APD}
∠
A
PB
=
∠
A
P
D
4
1
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Is it true ??
Is it true that for any permulation
a
1
,
a
2
.
.
.
.
.
,
a
2002
a_1,a_2.....,a_{2002}
a
1
,
a
2
.....
,
a
2002
of
1
,
2....
,
2002
1,2....,2002
1
,
2....
,
2002
there are positive integers
m
,
n
m,n
m
,
n
of the same parity such that
0
<
m
<
n
<
2003
0<m<n<2003
0
<
m
<
n
<
2003
and
a
m
+
a
n
=
2
a
m
+
n
2
a_m+a_n=2a_{\frac {m+n}{2}}
a
m
+
a
n
=
2
a
2
m
+
n
2
1
Hide problems
A funtional equation
Find all
f
:
R
−
R
f:R-R
f
:
R
−
R
such that
f
(
x
2
+
y
+
f
(
y
)
)
=
2
y
+
f
(
x
)
2
f(x^2+y+f(y))=2y+f(x)^2
f
(
x
2
+
y
+
f
(
y
))
=
2
y
+
f
(
x
)
2
1
1
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A combinatorial geometry
Cut
2003
2003
2003
disjoint rectangles from an acute-angled triangle
A
B
C
ABC
A
BC
, such that any of them has a parallel side to
A
B
AB
A
B
and the sum of their areas is maximal.
6
1
Hide problems
Solve !
In natural numbers
m
,
n
m,n
m
,
n
Solve :
n
(
n
+
1
)
(
n
+
2
)
(
n
+
3
)
=
m
(
m
+
1
)
2
(
m
+
2
)
3
(
m
+
3
)
4
n(n+1)(n+2)(n+3)=m(m+1)^2(m+2)^3(m+3)^4
n
(
n
+
1
)
(
n
+
2
)
(
n
+
3
)
=
m
(
m
+
1
)
2
(
m
+
2
)
3
(
m
+
3
)
4