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Contests
National and Regional Contests
China Contests
XES Mathematics Olympiad
the 12th XMO
Problem 5
Inequality
Inequality
Source: 12th XMO Test 1 Problem 7
April 14, 2023
inequalities
Problem Statement
Let
a
,
b
,
c
∈
R
+
a,b,c\in\mathbb R_+
a
,
b
,
c
∈
R
+
satisfy that
(
1
+
a
)
(
1
+
b
)
(
1
+
c
)
=
(
a
b
−
a
−
b
+
1
)
(
1
+
c
)
+
(
b
c
−
b
−
c
+
1
)
(
1
+
a
)
+
(
c
a
−
c
−
a
+
1
)
(
1
+
b
)
.
\sqrt{(1+a)(1+b)(1+c)}=\sqrt{(ab-a-b+1)(1+c)}+\sqrt{(bc-b-c+1)(1+a)}+\sqrt{(ca-c-a+1)(1+b)}.
(
1
+
a
)
(
1
+
b
)
(
1
+
c
)
=
(
ab
−
a
−
b
+
1
)
(
1
+
c
)
+
(
b
c
−
b
−
c
+
1
)
(
1
+
a
)
+
(
c
a
−
c
−
a
+
1
)
(
1
+
b
)
.
Find the value range of
a
+
b
+
c
.
a+b+c.
a
+
b
+
c
.
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