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Problems
Contests
National and Regional Contests
Serbia Contests
Serbia JBMO TST
2015 Junior Balkan Team Selection Test
2015 Junior Balkan Team Selection Test
Part of
Serbia JBMO TST
Subcontests
(4)
4
1
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Serbia Junior Balkan TST 2015
The diagonals
A
D
AD
A
D
,
B
E
BE
BE
,
C
F
CF
CF
of cyclic hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
intersect in
S
S
S
and
A
B
AB
A
B
is parallel to
C
F
CF
CF
and lines
D
E
DE
D
E
and
C
F
CF
CF
intersect each other in
M
M
M
. Let
N
N
N
be a point such that
M
M
M
is the midpoint of
S
N
SN
SN
. Prove that circumcircle of
△
A
D
N
\triangle ADN
△
A
D
N
is passing through midpoint of segment
C
F
CF
CF
.
3
1
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Serbia Junior Balkan TST 2015
Prove inequallity :
1
+
1
2
3
+
.
.
.
+
1
201
5
3
<
5
4
1+\frac{1}{2^3}+...+\frac{1}{2015^3}<\frac{5}{4}
1
+
2
3
1
+
...
+
201
5
3
1
<
4
5
2
1
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Serbia Junior Balkan TST 2015
Two different
3
3
3
digit numbers are picked and then for every of them is calculated sum of all
5
5
5
numbers which are getting when digits of picked number change place (etc. if one of the number is
707
707
707
, the sum is
2401
=
770
+
77
+
77
+
770
+
707
2401=770+77+77+770+707
2401
=
770
+
77
+
77
+
770
+
707
). Do the given results must be different?
1
1
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Serbia Junior Balkan TST 2015
Frog is in the origin of decartes coordinate system. Every second frog jumpes horizontally or vertically in some of the
4
4
4
adjacent points which coordinates are integers. Find number of different points in which frog can be found in
2015
2015
2015
seconds.