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Bundeswettbewerb Mathematik
1982 Bundeswettbewerb Mathematik
3
Bundeswettbewerb Mathematik 1982 Problem 2.3
Bundeswettbewerb Mathematik 1982 Problem 2.3
Source: Bundeswettbewerb Mathematik 1982 Round 2
September 22, 2022
algebra
inequalities
minimum
Problem Statement
Given that
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2, . . . , a_n
a
1
,
a
2
,
...
,
a
n
are nonnegative real numbers with
a
1
+
⋯
+
a
n
=
1
a_1 + \cdots + a_n = 1
a
1
+
⋯
+
a
n
=
1
, prove that the expression
a
1
1
+
a
2
+
a
3
+
⋯
+
a
n
+
a
2
1
+
a
1
+
a
3
+
⋯
+
a
n
+
⋯
+
a
n
1
+
a
1
+
a
2
+
⋯
+
a
n
−
1
\frac{a_1}{1+a_2 +a_3 +\cdots +a_n }\; +\; \frac{a_2}{1+a_1 +a_3 +\cdots +a_n }\; +\; \cdots \; +\, \frac{a_n }{1+a_1 +a_2+\cdots +a_{n-1} }
1
+
a
2
+
a
3
+
⋯
+
a
n
a
1
+
1
+
a
1
+
a
3
+
⋯
+
a
n
a
2
+
⋯
+
1
+
a
1
+
a
2
+
⋯
+
a
n
−
1
a
n
attains its minimum, and determine this minimum.
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