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Problems
Contests
International Contests
APMO
1989 APMO
1
1
Part of
1989 APMO
Problems
(1)
Inequality from the first apmo
Source: APMO 1989
3/10/2006
Let
x
1
x_1
x
1
,
x
2
x_2
x
2
,
⋯
\cdots
⋯
,
x
n
x_n
x
n
be positive real numbers, and let
S
=
x
1
+
x
2
+
⋯
+
x
n
.
S = x_1 + x_2 + \cdots + x_n.
S
=
x
1
+
x
2
+
⋯
+
x
n
.
Prove that
(
1
+
x
1
)
(
1
+
x
2
)
⋯
(
1
+
x
n
)
≤
1
+
S
+
S
2
2
!
+
S
3
3
!
+
⋯
+
S
n
n
!
(1 + x_1)(1 + x_2) \cdots (1 + x_n) \leq 1 + S + \frac{S^2}{2!} + \frac{S^3}{3!} + \cdots + \frac{S^n}{n!}
(
1
+
x
1
)
(
1
+
x
2
)
⋯
(
1
+
x
n
)
≤
1
+
S
+
2
!
S
2
+
3
!
S
3
+
⋯
+
n
!
S
n
inequalities
n-variable inequality