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National and Regional Contests
China Contests
China Team Selection Test
2023 China Team Selection Test
P11
2023 China TST Problem 11
2023 China TST Problem 11
Source: 2023 China TST Problem 11
March 18, 2023
inequalities
China TST
Probabilistic Method
Problem Statement
Let
n
∈
N
+
.
n\in\mathbb N_+.
n
∈
N
+
.
For
1
≤
i
,
j
,
k
≤
n
,
a
i
j
k
∈
{
−
1
,
1
}
.
1\leq i,j,k\leq n,a_{ijk}\in\{ -1,1\} .
1
≤
i
,
j
,
k
≤
n
,
a
ijk
∈
{
−
1
,
1
}
.
Prove that:
∃
x
1
,
x
2
,
⋯
,
x
n
,
y
1
,
y
2
,
⋯
,
y
n
,
z
1
,
z
2
,
⋯
,
z
n
∈
{
−
1
,
1
}
,
\exists x_1,x_2,\cdots ,x_n,y_1,y_2,\cdots ,y_n,z_1,z_2,\cdots ,z_n\in \{-1,1\} ,
∃
x
1
,
x
2
,
⋯
,
x
n
,
y
1
,
y
2
,
⋯
,
y
n
,
z
1
,
z
2
,
⋯
,
z
n
∈
{
−
1
,
1
}
,
satisfy
∣
∑
i
=
1
n
∑
j
=
1
n
∑
k
=
1
n
a
i
j
k
x
i
y
j
z
k
∣
>
n
2
3
.
\left| \sum\limits_{i=1}^n\sum\limits_{j=1}^n\sum\limits_{k=1}^na_{ijk}x_iy_jz_k\right| >\frac {n^2}3.
i
=
1
∑
n
j
=
1
∑
n
k
=
1
∑
n
a
ijk
x
i
y
j
z
k
>
3
n
2
.
Created by Yu Deng
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