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All-Russian Olympiad
1968 All Soviet Union Mathematical Olympiad
113
ASU 113 All Soviet Union MO 1968 |a_n|=|a_{n-1}+1|
ASU 113 All Soviet Union MO 1968 |a_n|=|a_{n-1}+1|
Source:
June 20, 2019
Sequence
algebra
inequalities
Problem Statement
The sequence
a
1
,
a
2
,
.
.
.
,
a
n
a_1,a_2,...,a_n
a
1
,
a
2
,
...
,
a
n
satisfies the following conditions:
a
1
=
0
,
∣
a
2
∣
=
∣
a
1
+
1
∣
,
.
.
.
,
∣
a
n
∣
=
∣
a
n
−
1
+
1
∣
.
a_1=0, |a_2|=|a_1+1|, ..., |a_n|=|a_{n-1}+1|.
a
1
=
0
,
∣
a
2
∣
=
∣
a
1
+
1∣
,
...
,
∣
a
n
∣
=
∣
a
n
−
1
+
1∣.
Prove that
(
a
1
+
a
2
+
.
.
.
+
a
n
)
/
n
≥
−
1
/
2
(a_1+a_2+...+a_n)/n \ge -1/2
(
a
1
+
a
2
+
...
+
a
n
)
/
n
≥
−
1/2
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