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Malaysia Contests
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JOM 2015 Shortlist
N2
N2
Part of
JOM 2015 Shortlist
Problems
(1)
Two sets
Source: Junior Olympiad of Malaysia Shortlist 2015 N2
7/17/2015
Let
A
⊂
N
\mathbb{A} \subset \mathbb{N}
A
⊂
N
such that all elements in
A
\mathbb{A}
A
can be representable in the form of
x
2
+
2
y
2
x^2+2y^2
x
2
+
2
y
2
,
x
,
y
∈
N
x,y \in \mathbb{N}
x
,
y
∈
N
, and
x
>
y
x>y
x
>
y
. Let
B
⊂
N
\mathbb{B} \subset \mathbb{N}
B
⊂
N
such that all elements in
B
\mathbb{B}
B
can be representable in the form of
a
3
+
b
3
+
c
3
a
+
b
+
c
\displaystyle \frac{a^3+b^3+c^3}{a+b+c}
a
+
b
+
c
a
3
+
b
3
+
c
3
,
a
,
b
,
c
∈
N
a,b,c \in \mathbb{N}
a
,
b
,
c
∈
N
, and
a
,
b
,
c
a,b,c
a
,
b
,
c
are distinct.a) Prove that
A
⊂
B
\mathbb{A} \subset \mathbb{B}
A
⊂
B
.b) Prove that there exist infinitely many positive integers
n
n
n
satisfy
n
∈
B
n \in \mathbb{B}
n
∈
B
and
n
∉
A
n \not \in \mathbb{A}
n
∈
A
number theory