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Problems
Contests
National and Regional Contests
Moldova Contests
JBMO TST - Moldova
2005 Junior Balkan Team Selection Tests - Moldova
6
6
Part of
2005 Junior Balkan Team Selection Tests - Moldova
Problems
(1)
min of x_1 +x_2^2/ 2 + x_3^3/ 3 +...+ x_n^n /n if sum 1/x_i=n
Source: 2005 Moldova JBMO TST p6
2/20/2021
Let
n
n
n
be a nonzero natural number, and
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2,..., x_n
x
1
,
x
2
,
...
,
x
n
positive real numbers that
1
x
1
+
1
x
2
+
.
.
.
+
1
x
n
=
n
\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}= n
x
1
1
+
x
2
1
+
...
+
x
n
1
=
n
. Find the minimum value of the expression
x
1
+
x
2
2
2
+
+
x
3
3
3
+
.
.
.
+
+
x
n
n
n
x_1 +\frac{x_2^2}{2}++\frac{x_3^3}{3}+...++\frac{x_n^n}{n}
x
1
+
2
x
2
2
+
+
3
x
3
3
+
...
+
+
n
x
n
n
.
algebra
inequalities