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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 3 of 4: Prove the cubic bound
Previous step
Next step
(
a
+
b
+
c
)
3
≥
a
3
+
b
3
+
c
3
+
24
a
b
c
.
(a+b+c)^3\ge a^3+b^3+c^3+24abc.
(
a
+
b
+
c
)
3
≥
a
3
+
b
3
+
c
3
+
24
ab
c
.
Detailed analysis
Expand and use (a+b+c)(ab+bc+ca) at least 9abc.
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