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National and Regional Contests
China Contests
China Team Selection Test
2023 China Team Selection Test
P11
P11
Part of
2023 China Team Selection Test
Problems
(1)
2023 China TST Problem 11
Source: 2023 China TST Problem 11
3/18/2023
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
inequalities
China TST
Probabilistic Method