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China Contests
China Western Mathematical Olympiad
2024 China Western Mathematical Olympiad
8
8
Part of
2024 China Western Mathematical Olympiad
Problems
(1)
Maximum value of n^2 reals in a grid
Source: 2024 CWMO P8
8/7/2024
Given a positive integer
n
≥
2
n \geq 2
n
≥
2
. Let
a
i
j
a_{ij}
a
ij
(
1
≤
i
,
j
≤
n
)
(1 \leq i,j \leq n)
(
1
≤
i
,
j
≤
n
)
be
n
2
n^2
n
2
non-negative reals and their sum is
1
1
1
. For
1
≤
i
≤
n
1\leq i \leq n
1
≤
i
≤
n
, define
R
i
=
m
a
x
1
≤
k
≤
n
(
a
i
k
)
R_i=max_{1\leq k \leq n}(a_{ik})
R
i
=
ma
x
1
≤
k
≤
n
(
a
ik
)
. For
1
≤
j
≤
n
1\leq j \leq n
1
≤
j
≤
n
, define
C
j
=
m
i
n
1
≤
k
≤
n
(
a
k
j
)
C_j=min_{1\leq k \leq n}(a_{kj})
C
j
=
mi
n
1
≤
k
≤
n
(
a
kj
)
Find the maximum value of
C
1
C
2
⋯
C
n
(
R
1
+
R
2
+
⋯
+
R
n
)
C_1C_2 \cdots C_n(R_1+R_2+ \cdots +R_n)
C
1
C
2
⋯
C
n
(
R
1
+
R
2
+
⋯
+
R
n
)
algebra
inequalities