MathDB
Problems
Contests
International Contests
IMO Longlists
1987 IMO Longlists
27
Find the smallest real number C
Find the smallest real number C
Source:
September 5, 2010
induction
inequalities
inequalities unsolved
Problem Statement
Find, with proof, the smallest real number
C
C
C
with the following property: For every infinite sequence
{
x
i
}
\{x_i\}
{
x
i
}
of positive real numbers such that
x
1
+
x
2
+
⋯
+
x
n
≤
x
n
+
1
x_1 + x_2 +\cdots + x_n \leq x_{n+1}
x
1
+
x
2
+
⋯
+
x
n
≤
x
n
+
1
for
n
=
1
,
2
,
3
,
⋯
n = 1, 2, 3, \cdots
n
=
1
,
2
,
3
,
⋯
, we have
x
1
+
x
2
+
⋯
+
x
n
≤
C
x
1
+
x
2
+
⋯
+
x
n
∀
n
∈
N
.
\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n} \leq C \sqrt{x_1+x_2+\cdots+x_n} \qquad \forall n \in \mathbb N.
x
1
+
x
2
+
⋯
+
x
n
≤
C
x
1
+
x
2
+
⋯
+
x
n
∀
n
∈
N
.
Back to Problems
View on AoPS