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National and Regional Contests
Serbia Contests
Serbia National Math Olympiad
2015 Serbia National Math Olympiad
4
Serbian TST 2015
Serbian TST 2015
Source:
May 25, 2015
number theory
Problem Statement
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.
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