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2023 Korea National Olympiad
2
x+2, x(x+1)/2
x+2, x(x+1)/2
Source: KMO 2023 P2
November 4, 2023
combinatorics
Problem Statement
Sets
A
0
,
A
1
,
…
,
A
2023
A_0, A_1, \dots, A_{2023}
A
0
,
A
1
,
…
,
A
2023
satisfy the following conditions:[*]
A
0
=
{
3
}
A_0 = \{ 3 \}
A
0
=
{
3
}
[*]
A
n
=
{
x
+
2
∣
x
∈
A
n
−
1
}
∪
{
x
(
x
+
1
)
/
2
∣
x
∈
A
n
−
1
}
A_n = \{ x + 2 \mid x \in A_{n - 1} \} \ \cup \{x(x+1) / 2 \mid x \in A_{n - 1} \}
A
n
=
{
x
+
2
∣
x
∈
A
n
−
1
}
∪
{
x
(
x
+
1
)
/2
∣
x
∈
A
n
−
1
}
for each
n
=
1
,
2
,
…
,
2023
n = 1, 2, \dots, 2023
n
=
1
,
2
,
…
,
2023
.Find
∣
A
2023
∣
|A_{2023}|
∣
A
2023
∣
.
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