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International Contests
Nordic
1999 Nordic
4
4
Part of
1999 Nordic
Problems
(1)
inequality with sums of inverses of positive reals
Source: Nordic Mathematical Contest 1999 #4
10/3/2017
Let
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2, . . . , a_n
a
1
,
a
2
,
...
,
a
n
be positive real numbers and
n
≥
1
n \ge 1
n
≥
1
. Show that
n
(
1
a
1
+
.
.
.
+
1
a
n
)
≥
(
1
1
+
a
1
+
.
.
.
+
1
1
+
a
n
)
(
n
+
1
a
1
+
.
.
.
+
1
a
n
)
n (\frac{1}{a_1}+...+\frac{1}{a_n}) \ge (\frac{1}{1+a_1}+...+\frac{1}{1+a_n})(n+\frac{1}{a_1}+...+\frac{1}{a_n})
n
(
a
1
1
+
...
+
a
n
1
)
≥
(
1
+
a
1
1
+
...
+
1
+
a
n
1
)
(
n
+
a
1
1
+
...
+
a
n
1
)
When does equality hold?
inequalities
positive