MathDB
Problems
Contests
National and Regional Contests
China Contests
China National Olympiad
2024 China National Olympiad
3
3
Part of
2024 China National Olympiad
Problems
(1)
Including chain of minimal sets under some valuation
Source: China MO 2024, Day 1, Problem 3
11/28/2023
Let
p
⩾
5
p \geqslant 5
p
⩾
5
be a prime and
S
=
{
1
,
2
,
…
,
p
}
S = \left\{ 1, 2, \ldots, p \right\}
S
=
{
1
,
2
,
…
,
p
}
. Define
r
(
x
,
y
)
r(x,y)
r
(
x
,
y
)
as follows:
r
(
x
,
y
)
=
{
y
−
x
y
⩾
x
y
−
x
+
p
y
<
x
.
r(x,y) = \begin{cases} y - x & y \geqslant x \\ y - x + p & y < x \end{cases}.
r
(
x
,
y
)
=
{
y
−
x
y
−
x
+
p
y
⩾
x
y
<
x
.
For a nonempty proper subset
A
A
A
of
S
S
S
, let
f
(
A
)
=
∑
x
∈
A
∑
y
∈
A
(
r
(
x
,
y
)
)
2
.
f(A) = \sum_{x \in A} \sum_{y \in A} \left( r(x,y) \right)^2.
f
(
A
)
=
x
∈
A
∑
y
∈
A
∑
(
r
(
x
,
y
)
)
2
.
A good subset of
S
S
S
is a nonempty proper subset
A
A
A
satisfying that for all subsets
B
⊆
S
B \subseteq S
B
⊆
S
of the same size as
A
A
A
,
f
(
B
)
⩾
f
(
A
)
f(B) \geqslant f(A)
f
(
B
)
⩾
f
(
A
)
. Find the largest integer
L
L
L
such that there exists distinct good subsets
A
1
⊆
A
2
⊆
…
⊆
A
L
A_1 \subseteq A_2 \subseteq \ldots \subseteq A_L
A
1
⊆
A
2
⊆
…
⊆
A
L
.Proposed by Bin Wang
combinatorics