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South Africa National Olympiad
2023 South Africa National Olympiad
4
4
Part of
2023 South Africa National Olympiad
Problems
(1)
Integers expressible as product of elements of an interesting set
Source: The South African Mathematical Olympiad P4
7/30/2023
Let
A
A
A
be a set of real numbers satisfying the following: (a)
(
n
2
+
1
)
∈
A
\sqrt(n^2+1) \in A
(
n
2
+
1
)
∈
A
for all positive integers
n
n
n
, (b) if
x
∈
A
x \in A
x
∈
A
and
y
∈
A
y \in A
y
∈
A
, then
x
−
y
∈
A
x-y \in A
x
−
y
∈
A
. Prove that every integer can be written as a product of two different elements in
A
A
A
.
number theory