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Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2020 Vietnam National Olympiad
7
7
Part of
2020 Vietnam National Olympiad
Problems
(1)
Set and graph
Source: VMO 2020, Day 2 - P7
12/28/2019
Given a positive integer
n
>
1
n>1
n
>
1
. Denote
T
T
T
a set that contains all ordered sets
(
x
;
y
;
z
)
(x;y;z)
(
x
;
y
;
z
)
such that
x
,
y
,
z
x,y,z
x
,
y
,
z
are all distinct positive integers and
1
≤
x
,
y
,
z
≤
2
n
1\leq x,y,z\leq 2n
1
≤
x
,
y
,
z
≤
2
n
. Also, a set
A
A
A
containing ordered sets
(
u
;
v
)
(u;v)
(
u
;
v
)
is called "connected" with
T
T
T
if for every
(
x
;
y
;
z
)
∈
T
(x;y;z)\in T
(
x
;
y
;
z
)
∈
T
then
{
(
x
;
y
)
,
(
x
;
z
)
,
(
y
;
z
)
}
∩
A
≠
∅
\{(x;y),(x;z),(y;z)\} \cap A \neq \varnothing
{(
x
;
y
)
,
(
x
;
z
)
,
(
y
;
z
)}
∩
A
=
∅
. a) Find the number of elements of set
T
T
T
. b) Prove that there exists a set "connected" with
T
T
T
that has exactly
2
n
(
n
−
1
)
2n(n-1)
2
n
(
n
−
1
)
elements. c) Prove that every set "connected" with
T
T
T
has at least
2
n
(
n
−
1
)
2n(n-1)
2
n
(
n
−
1
)
elements.
algebra
Sets