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Contests
National and Regional Contests
Serbia Contests
Serbia JBMO TST
2014 Serbia JBMO TST
2014 Serbia JBMO TST
Part of
Serbia JBMO TST
Subcontests
(4)
4
1
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There $100$ people seated at a round table $50$ women and $50$ men.
There
100
100
100
people seated at a round table
50
50
50
women and
50
50
50
men. Show that there are two people of opposite gender that stay between two people of opposite gender. (WWMM, MMWW, WMWM, MWMW)
1
1
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Inequality
For
a
,
b
,
c
,
d
,
e
a, b, c, d, e
a
,
b
,
c
,
d
,
e
in the interval
[
0
,
1
]
[0,1]
[
0
,
1
]
, prove that
(
1
+
a
+
b
+
c
+
d
+
e
)
2
=
>
4
(
a
2
+
b
2
+
c
2
+
d
2
+
e
2
)
(1+a+b+c+d+e)^2=>4(a^2+b^2+c^2+d^2+e^2)
(
1
+
a
+
b
+
c
+
d
+
e
)
2
=>
4
(
a
2
+
b
2
+
c
2
+
d
2
+
e
2
)
2
1
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Number theory ()
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be the natural numbers such that
2
a
+
4
b
+
5
c
=
201
4
d
.
2^a+4^b+5^c=2014^d.
2
a
+
4
b
+
5
c
=
201
4
d
.
Find all
(
a
,
b
,
c
,
d
)
.
(a,b,c,d).
(
a
,
b
,
c
,
d
)
.
3
1
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Nice geometry
Consider parallelogram
A
B
C
D
ABCD
A
BC
D
, with acute angle at
A
A
A
,
A
C
AC
A
C
and
B
D
BD
B
D
intersect at
E
E
E
. Circumscribed circle of triangle
A
C
D
ACD
A
C
D
intersects
A
B
AB
A
B
,
B
C
BC
BC
and
B
D
BD
B
D
at
K
K
K
,
L
L
L
and
P
P
P
(in that order). Then, circumscribed circle of triangle
C
E
L
CEL
CE
L
intersects
B
D
BD
B
D
at
M
M
M
. Prove that:
K
D
∗
K
M
=
K
L
∗
P
C
KD*KM=KL*PC
KD
∗
K
M
=
K
L
∗
PC