MathDB
Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgarian Winter Tournament
2024 Bulgarian Winter Tournament
12.3
12.3
Part of
2024 Bulgarian Winter Tournament
Problems
(1)
Probabilistic sum is strictly increasing
Source: Bulgarian Winter Tournament 2024 12.3
1/28/2024
Let
n
n
n
be a positive integer and let
A
\mathcal{A}
A
be a family of non-empty subsets of
{
1
,
2
,
…
,
n
}
\{1, 2, \ldots, n \}
{
1
,
2
,
…
,
n
}
such that if
A
∈
A
A \in \mathcal{A}
A
∈
A
and
A
A
A
is subset of a set
B
⊆
{
1
,
2
,
…
,
n
}
B\subseteq \{1, 2, \ldots, n\}
B
⊆
{
1
,
2
,
…
,
n
}
, then
B
B
B
is also in
A
\mathcal{A}
A
. Show that the function
f
(
x
)
:
=
∑
A
∈
A
x
∣
A
∣
(
1
−
x
)
n
−
∣
A
∣
f(x):=\sum_{A \in \mathcal{A}} x^{|A|}(1-x)^{n-|A|}
f
(
x
)
:=
A
∈
A
∑
x
∣
A
∣
(
1
−
x
)
n
−
∣
A
∣
is strictly increasing for
x
∈
(
0
,
1
)
x \in (0,1)
x
∈
(
0
,
1
)
.
combinatorics
probability