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1974 IMO Shortlist
9
Prove the identity if x+y+z=xyz
Prove the identity if x+y+z=xyz
Source:
September 22, 2010
trigonometry
algebra
Trigonometric Identities
trigonometric substitution
equation
IMO Shortlist
Problem Statement
Let
x
,
y
,
z
x, y, z
x
,
y
,
z
be real numbers each of whose absolute value is different from
1
3
\frac{1}{\sqrt 3}
3
1
such that
x
+
y
+
z
=
x
y
z
x + y + z = xyz
x
+
y
+
z
=
x
yz
. Prove that
3
x
−
x
3
1
−
3
x
2
+
3
y
−
y
3
1
−
3
y
2
+
3
z
−
z
3
1
−
3
z
2
=
3
x
−
x
3
1
−
3
x
2
⋅
3
y
−
y
3
1
−
3
y
2
⋅
3
z
−
z
3
1
−
3
z
2
\frac{3x - x^3}{1-3x^2} + \frac{3y - y^3}{1-3y^2} + \frac{3z -z^3}{1-3z^2} = \frac{3x - x^3}{1-3x^2} \cdot \frac{3y - y^3}{1-3y^2} \cdot \frac{3z - z^3}{1-3z^2}
1
−
3
x
2
3
x
−
x
3
+
1
−
3
y
2
3
y
−
y
3
+
1
−
3
z
2
3
z
−
z
3
=
1
−
3
x
2
3
x
−
x
3
⋅
1
−
3
y
2
3
y
−
y
3
⋅
1
−
3
z
2
3
z
−
z
3
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