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Problems
Contests
National and Regional Contests
Moldova Contests
JBMO TST - Moldova
2005 Junior Balkan Team Selection Tests - Moldova
3
Inequality
Inequality
Source: Moldova JTST 2005
March 1, 2018
inequalities
Problem Statement
Let
a
1
,
a
2
,
.
.
.
a
n
a_1,a_2,...a_n
a
1
,
a
2
,
...
a
n
be positive numbers. And let
s
=
a
1
+
a
2
+
.
.
.
+
a
n
s=a_1+a_2+...+a_n
s
=
a
1
+
a
2
+
...
+
a
n
,and
p
=
a
1
∗
a
2
∗
.
.
.
∗
a
n
p=a_1*a_2*...*a_n
p
=
a
1
∗
a
2
∗
...
∗
a
n
.Prove that
2
n
∗
p
≤
1
+
s
1
!
+
s
2
2
!
+
.
.
.
+
s
n
n
!
2^{n}*\sqrt{p} \leq 1+\frac{s}{1!}+\frac{s^2}{2!}+...+\frac{s^n}{n!}
2
n
∗
p
≤
1
+
1
!
s
+
2
!
s
2
+
...
+
n
!
s
n
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