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Problems
Contests
National and Regional Contests
China Contests
South East Mathematical Olympiad
2020 South East Mathematical Olympiad
5
5
Part of
2020 South East Mathematical Olympiad
Problems
(1)
Exhausting number theory
Source: 2020 China Southeast 10.5/11.5
8/9/2020
Consider the set
I
=
{
1
,
2
,
⋯
,
2020
}
I=\{ 1,2, \cdots, 2020 \}
I
=
{
1
,
2
,
⋯
,
2020
}
. Let
W
=
{
w
(
a
,
b
)
=
(
a
+
b
)
+
a
b
∣
a
,
b
∈
I
}
∩
I
W= \{w(a,b)=(a+b)+ab | a,b \in I \} \cap I
W
=
{
w
(
a
,
b
)
=
(
a
+
b
)
+
ab
∣
a
,
b
∈
I
}
∩
I
,
Y
=
{
y
(
a
,
b
)
=
(
a
+
b
)
⋅
a
b
∣
a
,
b
∈
I
}
∩
I
Y=\{y(a,b)=(a+b) \cdot ab | a,b \in I \} \cap I
Y
=
{
y
(
a
,
b
)
=
(
a
+
b
)
⋅
ab
∣
a
,
b
∈
I
}
∩
I
be its two subsets. Further, let
X
=
W
∩
Y
X= W \cap Y
X
=
W
∩
Y
. (1) Find the sum of maximal and minimal elements in
X
X
X
. (2) An element
n
=
y
(
a
,
b
)
(
a
≤
b
)
n=y(a,b) (a \le b)
n
=
y
(
a
,
b
)
(
a
≤
b
)
in
Y
Y
Y
is called excellent, if its representation is not unique (for instance,
20
=
y
(
1
,
5
)
=
y
(
2
,
3
)
20=y(1,5)=y(2,3)
20
=
y
(
1
,
5
)
=
y
(
2
,
3
)
). Find the number of excellent elements in
Y
Y
Y
.(2) is only for Grade 11.
number theory