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International Contests
APMO
2007 APMO
4
4
Part of
2007 APMO
Problems
(1)
$\sqrt{x} + \sqrt{y} + \sqrt{z} = 1$.
Source: APMO 2007
3/31/2007
Let
x
;
y
x; y
x
;
y
and
z
z
z
be positive real numbers such that
x
+
y
+
z
=
1
\sqrt{x}+\sqrt{y}+\sqrt{z}= 1
x
+
y
+
z
=
1
. Prove that
x
2
+
y
z
2
x
2
(
y
+
z
)
+
y
2
+
z
x
2
y
2
(
z
+
x
)
+
z
2
+
x
y
2
z
2
(
x
+
y
)
≥
1.
\frac{x^{2}+yz}{\sqrt{2x^{2}(y+z)}}+\frac{y^{2}+zx}{\sqrt{2y^{2}(z+x)}}+\frac{z^{2}+xy}{\sqrt{2z^{2}(x+y)}}\geq 1.
2
x
2
(
y
+
z
)
x
2
+
yz
+
2
y
2
(
z
+
x
)
y
2
+
z
x
+
2
z
2
(
x
+
y
)
z
2
+
x
y
≥
1.
inequalities