MathDB
Problems
Contests
National and Regional Contests
China Contests
China National Olympiad
2024 China National Olympiad
4
4
Part of
2024 China National Olympiad
Problems
(1)
Find maximum number of pairs whose product is at least 1
Source: 2024 China MO, Day 2, Problem 4
11/29/2023
Let
a
1
,
a
2
,
…
,
a
2023
a_1, a_2, \ldots, a_{2023}
a
1
,
a
2
,
…
,
a
2023
be nonnegative real numbers such that
a
1
+
a
2
+
…
+
a
2023
=
100
a_1 + a_2 + \ldots + a_{2023} = 100
a
1
+
a
2
+
…
+
a
2023
=
100
. Let
A
=
{
(
i
,
j
)
∣
1
⩽
i
⩽
j
⩽
2023
,
a
i
a
j
⩾
1
}
A = \left \{ (i,j) \mid 1 \leqslant i \leqslant j \leqslant 2023, \, a_ia_j \geqslant 1 \right\}
A
=
{
(
i
,
j
)
∣
1
⩽
i
⩽
j
⩽
2023
,
a
i
a
j
⩾
1
}
. Prove that
∣
A
∣
⩽
5050
|A| \leqslant 5050
∣
A
∣
⩽
5050
and determine when the equality holds.Proposed by Yunhao Fu
algebra