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National and Regional Contests
Russia Contests
Soros Olympiad in Mathematics
I Soros Olympiad 1994-95 (Rus + Ukr)
10.5
| a_1+a_2+...+a_n - nS | <= 997 (I Soros Olympiad 1994-95 R1 10.5)
| a_1+a_2+...+a_n - nS | <= 997 (I Soros Olympiad 1994-95 R1 10.5)
Source:
July 31, 2021
algebra
inequalities
Problem Statement
Let
a
1
,
a
2
,
.
.
.
,
a
1994
a_1,a_2,...,a_{1994}
a
1
,
a
2
,
...
,
a
1994
be real numbers in the interval
[
−
1
,
1
]
[-1,1]
[
−
1
,
1
]
,
S
=
a
1
+
a
2
+
.
.
.
+
a
1994
1994
.
S=\frac{a_1+a_2+...+a_{1994}}{1994}.
S
=
1994
a
1
+
a
2
+
...
+
a
1994
.
Prove that for an arbitrary natural ,
1
≤
n
≤
1994
1\le n \le 1994
1
≤
n
≤
1994
, holds the inequality
∣
a
1
+
a
2
+
.
.
.
+
a
n
−
n
S
∣
≤
997.
| a_1+a_2+...+a_n - nS | \le 997.
∣
a
1
+
a
2
+
...
+
a
n
−
n
S
∣
≤
997.
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