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2019 Turkey MO (2nd round)
5
5
Part of
2019 Turkey MO (2nd round)
Problems
(1)
Binary-valued function
Source: Turkey National Mathematical Olympiad 2019, Problem 5
12/23/2019
Let
f
:
{
1
,
2
,
…
,
2019
}
→
{
−
1
,
1
}
f:\{1,2,\dots,2019\}\to\{-1,1\}
f
:
{
1
,
2
,
…
,
2019
}
→
{
−
1
,
1
}
be a function, such that for every
k
∈
{
1
,
2
,
…
,
2019
}
k\in\{1,2,\dots,2019\}
k
∈
{
1
,
2
,
…
,
2019
}
, there exists an
ℓ
∈
{
1
,
2
,
…
,
2019
}
\ell\in\{1,2,\dots,2019\}
ℓ
∈
{
1
,
2
,
…
,
2019
}
such that
∑
i
∈
Z
:
(
ℓ
−
i
)
(
i
−
k
)
⩾
0
f
(
i
)
⩽
0.
\sum_{i\in\mathbb{Z}:(\ell-i)(i-k)\geqslant 0} f(i)\leqslant 0.
i
∈
Z
:
(
ℓ
−
i
)
(
i
−
k
)
⩾
0
∑
f
(
i
)
⩽
0.
Determine the maximum possible value of
∑
i
∈
Z
:
1
⩽
i
⩽
2019
f
(
i
)
.
\sum_{i\in\mathbb{Z}:1\leqslant i\leqslant 2019} f(i).
i
∈
Z
:
1
⩽
i
⩽
2019
∑
f
(
i
)
.
function
algebra
combinatorics