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National and Regional Contests
Thailand Contests
Thailand National Olympiad
2019 Thailand Mathematical Olympiad
7
7
Part of
2019 Thailand Mathematical Olympiad
Problems
(1)
Condition to maximize an expression
Source: 2019 Thailand Mathematical Olympiad P7
5/22/2019
Let
A
=
{
−
2562
,
−
2561
,
.
.
.
,
2561
,
2562
}
A=\{-2562,-2561,...,2561,2562\}
A
=
{
−
2562
,
−
2561
,
...
,
2561
,
2562
}
. Prove that for any bijection (1-1, onto function)
f
:
A
→
A
f:A\to A
f
:
A
→
A
,
∑
k
=
1
2562
∣
f
(
k
)
−
f
(
−
k
)
∣
is maximized if and only if
f
(
k
)
f
(
−
k
)
<
0
for any
k
=
1
,
2
,
.
.
.
,
2562.
\sum_{k=1}^{2562}\left\lvert f(k)-f(-k)\right\rvert\text{ is maximized if and only if } f(k)f(-k)<0\text{ for any } k=1,2,...,2562.
k
=
1
∑
2562
∣
f
(
k
)
−
f
(
−
k
)
∣
is maximized if and only if
f
(
k
)
f
(
−
k
)
<
0
for any
k
=
1
,
2
,
...
,
2562.
combinatorics