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IMO LongList 1985 BEL1 - Prove the equation
IMO LongList 1985 BEL1 - Prove the equation
Source:
September 10, 2010
number theory
least common multiple
algebra proposed
algebra
Problem Statement
Let
x
,
y
x, y
x
,
y
, and
z
z
z
be real numbers satisfying
x
+
y
+
z
=
x
y
z
.
x + y + z = xyz.
x
+
y
+
z
=
x
yz
.
Prove that
x
(
1
−
y
2
)
(
1
−
z
2
)
+
y
(
1
−
z
2
)
(
1
−
x
2
)
+
z
(
1
−
x
2
)
(
1
−
y
2
)
=
4
x
y
z
.
x(1 - y^2)(1 - z^2) + y(1 -z^2)(1 - x^2) + z(1 - x^2)(1 - y^2) = 4xyz.
x
(
1
−
y
2
)
(
1
−
z
2
)
+
y
(
1
−
z
2
)
(
1
−
x
2
)
+
z
(
1
−
x
2
)
(
1
−
y
2
)
=
4
x
yz
.
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