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Nordic
2012 Nordic
1
This expression is an integer
This expression is an integer
Source: Nordic MO 2012 Q1
April 21, 2013
algebra unsolved
algebra
Problem Statement
The real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
are such that
a
2
+
b
2
=
2
c
2
a^2 + b^2 = 2c^2
a
2
+
b
2
=
2
c
2
, and also such that
a
≠
b
,
c
≠
−
a
,
c
≠
−
b
a \ne b, c \ne -a, c \ne -b
a
=
b
,
c
=
−
a
,
c
=
−
b
. Show that
(
a
+
b
+
2
c
)
(
2
a
2
−
b
2
−
c
2
)
(
a
−
b
)
(
a
+
c
)
(
b
+
c
)
\frac{(a+b+2c)(2a^2-b^2-c^2)}{(a-b)(a+c)(b+c)}
(
a
−
b
)
(
a
+
c
)
(
b
+
c
)
(
a
+
b
+
2
c
)
(
2
a
2
−
b
2
−
c
2
)
is an integer.
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