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2000 239 Open Mathematical Olympiad
3
$\frac{a_1 + a_2}{2}$ $\frac{a_1+ a_2 +a_3}{2 \sqrt{2}}$
$\frac{a_1 + a_2}{2}$ $\frac{a_1+ a_2 +a_3}{2 \sqrt{2}}$
Source: 239 2000 S3
May 18, 2020
Inequality
algebra
Problem Statement
For all positive real numbers
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \dots, a_n
a
1
,
a
2
,
…
,
a
n
, prove that
a
1
+
a
2
2
⋅
a
2
+
a
3
2
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⋯
⋅
a
n
+
a
1
2
≤
a
1
+
a
2
+
a
3
2
2
⋅
a
2
+
a
3
+
a
4
2
2
⋅
⋯
⋅
a
n
+
a
1
+
a
2
2
2
.
\frac{a_1\! +\! a_2}{2} \cdot \frac{a_2\! +\! a_3}{2} \cdot \dots \cdot \frac{a_n\! +\! a_1}{2} \leq \frac{a_1\!+\!a_2\!+\!a_3}{2 \sqrt{2}} \cdot \frac{a_2\!+\!a_3\!+\!a_4}{2 \sqrt{2}} \cdot \dots \cdot \frac{a_n\!+\!a_1\!+\!a_2}{2 \sqrt{2}}.
2
a
1
+
a
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⋅
2
a
2
+
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3
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⋯
⋅
2
a
n
+
a
1
≤
2
2
a
1
+
a
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+
a
3
⋅
2
2
a
2
+
a
3
+
a
4
⋅
⋯
⋅
2
2
a
n
+
a
1
+
a
2
.
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