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National and Regional Contests
Korea Contests
Korea National Olympiad
2014 Korea National Olympiad
1
square number
square number
Source: Korea National 2014 #1
January 21, 2015
number theory
greatest common divisor
number theory proposed
Problem Statement
For
x
,
y
x, y
x
,
y
positive integers,
x
2
−
4
y
+
1
x^2-4y+1
x
2
−
4
y
+
1
is a multiple of
(
x
−
2
y
)
(
1
−
2
y
)
(x-2y)(1-2y)
(
x
−
2
y
)
(
1
−
2
y
)
. Prove that
∣
x
−
2
y
∣
|x-2y|
∣
x
−
2
y
∣
is a square number.
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