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Contests
National and Regional Contests
Serbia Contests
Serbia National Math Olympiad
2015 Serbia National Math Olympiad
2015 Serbia National Math Olympiad
Part of
Serbia National Math Olympiad
Subcontests
(6)
6
1
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Serbian TST 2015
In nonnegative set of integers solve the equation:
(
2
2015
+
1
)
x
+
2
2015
=
2
y
+
1
(2^{2015}+1)^x + 2^{2015}=2^y+1
(
2
2015
+
1
)
x
+
2
2015
=
2
y
+
1
4
1
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Serbian TST 2015
For integer
a
a
a
,
a
≠
0
a \neq 0
a
=
0
,
v
2
(
a
)
v_2(a)
v
2
(
a
)
is greatest nonnegative integer
k
k
k
such that
2
k
∣
a
2^k | a
2
k
∣
a
. For given
n
∈
N
n \in \mathbb{N}
n
∈
N
determine highest possible cardinality of subset
A
A
A
of set
{
1
,
2
,
3
,
.
.
.
,
2
n
}
\{1,2,3,...,2^n \}
{
1
,
2
,
3
,
...
,
2
n
}
with following property: For all
x
,
y
∈
A
x, y \in A
x
,
y
∈
A
,
x
≠
y
x \neq y
x
=
y
, number
v
2
(
x
−
y
)
v_2(x-y)
v
2
(
x
−
y
)
is even.
5
1
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Serbia national Olympiad Day 2 Problem 2
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be nonnegative positive integers. Prove
x
−
y
x
y
+
2
y
+
1
+
y
−
z
z
y
+
2
z
+
1
+
z
−
x
x
z
+
2
x
+
1
≥
0
\frac{x-y}{xy+2y+1}+\frac{y-z}{zy+2z+1}+\frac{z-x}{xz+2x+1}\ge 0
x
y
+
2
y
+
1
x
−
y
+
zy
+
2
z
+
1
y
−
z
+
x
z
+
2
x
+
1
z
−
x
≥
0
2
1
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Serbia national Olympiad Day 1 problem 2
Let
k
k
k
be fixed positive integer . Let
F
k
(
n
)
Fk(n)
F
k
(
n
)
be smallest positive integer bigger than
k
n
kn
kn
such that
F
k
(
n
)
∗
n
Fk(n)*n
F
k
(
n
)
∗
n
is a perfect square . Prove that if
F
k
(
n
)
=
F
k
(
m
)
Fk(n)=Fk(m)
F
k
(
n
)
=
F
k
(
m
)
than
m
=
n
m=n
m
=
n
.
1
1
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Serbia National Olympiad Day 1 Problem 1
Consider circle inscribed quadriateral
A
B
C
D
ABCD
A
BC
D
. Let
M
,
N
,
P
,
Q
M,N,P,Q
M
,
N
,
P
,
Q
be midpoints of sides
D
A
,
A
B
,
B
C
,
C
D
DA,AB,BC,CD
D
A
,
A
B
,
BC
,
C
D
.Let
E
E
E
be the point of intersection of diagonals. Let
k
1
,
k
2
k1,k2
k
1
,
k
2
be circles around
E
M
N
EMN
EMN
and
E
P
Q
EPQ
EPQ
. Let
F
F
F
be point of intersection of
k
1
k1
k
1
and
k
2
k2
k
2
different from
E
E
E
. Prove that
E
F
EF
EF
is perpendicular to
A
C
AC
A
C
.
3
1
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Prisioners
We have
2015
2015
2015
prisinoers.The king gives everyone a hat coloured in one of
5
5
5
colors.Everyone sees all hats expect his own.Now,the King orders them in a line(a prisioner can see all guys behind and in front of him).The king asks the prisinoers one by one does he know the color of his hat.If he answers NO,then he is killed.If he answers YES,then answers which color is his hat,if his answers is true,he goes to freedom,if not,he is killed.All the prisinors can hear did he answer YES or NO,but if he answered YES,they don't know what did he answered(he is killed in public).They can think of a strategy before the King comes,but after that they can't comunicate.What is the largest number of prisinors we can guarentee that can survive?