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Ireland Contests
Ireland National Math Olympiad
2019 Irish Math Olympiad
9
9
Part of
2019 Irish Math Olympiad
Problems
(1)
solve 8xyz <= 1 and 8xyz = 1 if x^2 + y^2 + z^2 + 2xyz = 1$
Source: 2019 Irish Mathematical Olympiad paper 2 p9
10/5/2020
Suppose
x
,
y
,
z
x, y, z
x
,
y
,
z
are real numbers such that
x
2
+
y
2
+
z
2
+
2
x
y
z
=
1
x^2 + y^2 + z^2 + 2xyz = 1
x
2
+
y
2
+
z
2
+
2
x
yz
=
1
. Prove that
8
x
y
z
≤
1
8xyz \le 1
8
x
yz
≤
1
, with equality if and only if
(
x
,
y
,
z
)
(x, y,z)
(
x
,
y
,
z
)
is one of the following:
(
1
2
,
1
2
,
1
2
)
,
(
−
1
2
,
−
1
2
,
1
2
)
,
(
−
1
2
,
1
2
,
−
1
2
)
,
(
1
2
,
−
1
2
,
−
1
2
)
\left( \frac12, \frac12, \frac12 \right) , \left( -\frac12, -\frac12, \frac12 \right), \left(- \frac12, \frac12, -\frac12 \right), \left( \frac12,- \frac12, - \frac12 \right)
(
2
1
,
2
1
,
2
1
)
,
(
−
2
1
,
−
2
1
,
2
1
)
,
(
−
2
1
,
2
1
,
−
2
1
)
,
(
2
1
,
−
2
1
,
−
2
1
)
algebra
inequalities