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National and Regional Contests
China Contests
China Team Selection Test
2018 China Team Selection Test
6
The set of 3-Tuple Subscript
The set of 3-Tuple Subscript
Source: 2018 China TST 4 Day 2 Problem 6
March 27, 2018
combinatorics
algebraic combinatorics
algebra
combinatorics unsolved
Problem Statement
Suppose
a
i
,
b
i
,
c
i
,
i
=
1
,
2
,
⋯
,
n
a_i, b_i, c_i, i=1,2,\cdots ,n
a
i
,
b
i
,
c
i
,
i
=
1
,
2
,
⋯
,
n
, are
3
n
3n
3
n
real numbers in the interval
[
0
,
1
]
.
\left [ 0,1 \right ].
[
0
,
1
]
.
Define
S
=
{
(
i
,
j
,
k
)
∣
a
i
+
b
j
+
c
k
<
1
}
,
T
=
{
(
i
,
j
,
k
)
∣
a
i
+
b
j
+
c
k
>
2
}
.
S=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k<1 \right \}, \; \; T=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k>2 \right \}.
S
=
{
(
i
,
j
,
k
)
∣
a
i
+
b
j
+
c
k
<
1
}
,
T
=
{
(
i
,
j
,
k
)
∣
a
i
+
b
j
+
c
k
>
2
}
.
Now we know that
∣
S
∣
≥
2018
,
∣
T
∣
≥
2018.
\left | S \right |\ge 2018,\, \left | T \right |\ge 2018.
∣
S
∣
≥
2018
,
∣
T
∣
≥
2018.
Try to find the minimal possible value of
n
n
n
.
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