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Soros Olympiad in Mathematics
VI Soros Olympiad 1999 - 2000 (Russia)
10.8
product a_i >= 99^{100} - VI Soros Olympiad 1999-00 Round 1 10.8
product a_i >= 99^{100} - VI Soros Olympiad 1999-00 Round 1 10.8
Source:
May 21, 2024
inequalities
algebra
Problem Statement
There are
100
100
100
positive numbers
a
1
a_1
a
1
,
a
2
a_2
a
2
,
.
.
.
...
...
,
a
100
a_{100}
a
100
such that
1
a
1
+
1
+
1
a
2
+
1
+
.
.
.
+
1
a
100
+
1
≤
1.
\frac{1}{a_1+1}+\frac{1}{a_2+1}+...+\frac{1}{a_{100}+1} \le 1.
a
1
+
1
1
+
a
2
+
1
1
+
...
+
a
100
+
1
1
≤
1.
Prove that
a
1
⋅
a
2
⋅
.
.
.
⋅
a
100
≥
9
9
100
.
a_1 \cdot a_2\cdot ... \cdot a_{100} \ge 99^{100}.
a
1
⋅
a
2
⋅
...
⋅
a
100
≥
9
9
100
.
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