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International Mathematical Olympiad
›
2001
›
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 2 of 4: Expand
Previous step
Next step
∑
c
y
c
a
(
a
2
+
8
b
c
)
=
a
3
+
b
3
+
c
3
+
24
a
b
c
.
\sum_{cyc}a(a^2+8bc)=a^3+b^3+c^3+24abc.
cy
c
∑
a
(
a
2
+
8
b
c
)
=
a
3
+
b
3
+
c
3
+
24
ab
c
.
Detailed analysis
Collect the cyclic terms.
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