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National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2000 Moldova Team Selection Test
11
11
Part of
2000 Moldova Team Selection Test
Problems
(1)
$$\sum_{A\in M} (-1)^{n-|A|}\cdot f(A)=f(S)-\max\{f(A)|A\in M, A\neq S\},$$
Source: Moldova TST 2000
8/7/2023
Let
S
S
S
be a finite set with
n
n{}
n
(
n
>
1
)
(n>1)
(
n
>
1
)
elements,
M
M{}
M
the set of all subsets of
S
S{}
S
and a function
f
:
M
→
R
f:M\rightarrow\mathbb{R}
f
:
M
→
R
, that verifies the relation
f
(
A
∩
B
)
=
min
{
f
(
A
)
,
f
(
B
)
}
,
∀
A
,
B
∈
M
f(A\cap B)=\min\{f(A),f(B)\}, \forall A,B\in M
f
(
A
∩
B
)
=
min
{
f
(
A
)
,
f
(
B
)}
,
∀
A
,
B
∈
M
. Show that
∑
A
∈
M
(
−
1
)
n
−
∣
A
∣
⋅
f
(
A
)
=
f
(
S
)
−
max
{
f
(
A
)
∣
A
∈
M
,
A
≠
S
}
,
\sum_{A\in M} (-1)^{n-|A|}\cdot f(A)=f(S)-\max\{f(A)|A\in M, A\neq S\},
A
∈
M
∑
(
−
1
)
n
−
∣
A
∣
⋅
f
(
A
)
=
f
(
S
)
−
max
{
f
(
A
)
∣
A
∈
M
,
A
=
S
}
,
where
∣
A
∣
|A|
∣
A
∣
is the number of elements of subset
A
A{}
A
.
function