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2006 China National Olympiad
3
Equation with mn=k^2+k+3
Equation with mn=k^2+k+3
Source: CMO 2006
January 12, 2006
quadratics
number theory unsolved
number theory
Problem Statement
Positive integers
k
,
m
,
n
k, m, n
k
,
m
,
n
satisfy
m
n
=
k
2
+
k
+
3
mn=k^2+k+3
mn
=
k
2
+
k
+
3
, prove that at least one of the equations
x
2
+
11
y
2
=
4
m
x^2+11y^2=4m
x
2
+
11
y
2
=
4
m
and
x
2
+
11
y
2
=
4
n
x^2+11y^2=4n
x
2
+
11
y
2
=
4
n
has an odd solution.
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