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1987 IMO Longlists
76
76
Part of
1987 IMO Longlists
Problems
(1)
IMO LongList 1987 - prove the existence of limit
Source:
9/6/2010
Given two sequences of positive numbers
{
a
k
}
\{a_k\}
{
a
k
}
and
{
b
k
}
(
k
∈
N
)
\{b_k\} \ (k \in \mathbb N)
{
b
k
}
(
k
∈
N
)
such that:(i)
a
k
<
b
k
,
a_k < b_k,
a
k
<
b
k
,
(ii)
cos
a
k
x
+
cos
b
k
x
≥
−
1
k
\cos a_kx + \cos b_kx \geq -\frac 1k
cos
a
k
x
+
cos
b
k
x
≥
−
k
1
for all
k
∈
N
k \in \mathbb N
k
∈
N
and
x
∈
R
,
x \in \mathbb R,
x
∈
R
,
prove the existence of
lim
k
→
∞
a
k
b
k
\lim_{k \to \infty} \frac{a_k}{b_k}
lim
k
→
∞
b
k
a
k
and find this limit.
trigonometry
limit
algebra proposed
algebra