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International Mathematical Olympiad
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2000
›
Problem 2
Problem 2
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers with
a
b
c
=
1
abc=1
ab
c
=
1
. Prove that
(
a
−
1
+
1
b
)
(
b
−
1
+
1
c
)
(
c
−
1
+
1
a
)
≤
1
(a-1+\frac1b)(b-1+\frac1c)(c-1+\frac1a)\le 1
(
a
−
1
+
b
1
)
(
b
−
1
+
c
1
)
(
c
−
1
+
a
1
)
≤
1
.
Step 2 of 4: Rewrite
Previous step
Next step
(
x
−
y
+
z
)
(
x
+
y
−
z
)
(
−
x
+
y
+
z
)
x
y
z
.
\frac{(x-y+z)(x+y-z)(-x+y+z)}{xyz}.
x
y
z
(
x
−
y
+
z
)
(
x
+
y
−
z
)
(
−
x
+
y
+
z
)
.
Detailed analysis
Substitution gives this product.
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