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Hard Problem Involving Modulus
Hard Problem Involving Modulus
Source:
December 9, 2010
induction
algebra unsolved
algebra
Problem Statement
Let
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \ldots, a_n
a
1
,
a
2
,
…
,
a
n
be real numbers lying in
[
−
1
,
1
]
[-1, 1]
[
−
1
,
1
]
such that
a
1
+
a
2
+
⋯
+
a
n
=
0
a_1 + a_2 + \cdots + a_n = 0
a
1
+
a
2
+
⋯
+
a
n
=
0
. Prove that there is a
k
∈
{
1
,
2
,
…
,
n
}
k \in \{1, 2, \ldots, n\}
k
∈
{
1
,
2
,
…
,
n
}
such that
∣
a
1
+
2
a
2
+
3
a
3
+
⋯
+
k
a
k
∣
≤
2
k
+
1
4
|a_1 + 2a_2 + 3a_3 + \cdots + k a_k | \le \frac{2k+1}4
∣
a
1
+
2
a
2
+
3
a
3
+
⋯
+
k
a
k
∣
≤
4
2
k
+
1
.
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