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International Mathematical Olympiad
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2001
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Problem 2
Problem 2
For all positive real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
, prove
a
a
2
+
8
b
c
+
b
b
2
+
8
c
a
+
c
c
2
+
8
a
b
≥
1
\frac{a}{\sqrt{a^2+8bc}}+\frac{b}{\sqrt{b^2+8ca}}+\frac{c}{\sqrt{c^2+8ab}}\ge1
a
2
+
8
b
c
a
+
b
2
+
8
c
a
b
+
c
2
+
8
ab
c
≥
1
.
Step 1 of 4: Apply Hölder
Previous step
Next step
(
∑
c
y
c
a
a
2
+
8
b
c
)
2
(
∑
c
y
c
a
(
a
2
+
8
b
c
)
)
≥
(
a
+
b
+
c
)
3
.
(\sum_{cyc}\frac{a}{\sqrt{a^2+8bc}})^2(\sum_{cyc}a(a^2+8bc))\ge(a+b+c)^3.
(
cy
c
∑
a
2
+
8
b
c
a
)
2
(
cy
c
∑
a
(
a
2
+
8
b
c
))
≥
(
a
+
b
+
c
)
3
.
Detailed analysis
Use the weighted Hölder inequality.
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