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Germany Contests
German National Olympiad
2020 German National Olympiad
5
An inequality with integer variables
An inequality with integer variables
Source: Germany 2020, Problem 5
June 22, 2020
inequalities
inequalities proposed
Integers
Problem Statement
Let
a
1
,
a
2
,
…
,
a
22
a_1,a_2,\dots,a_{22}
a
1
,
a
2
,
…
,
a
22
be positive integers with sum
59
59
59
. Prove the inequality
a
1
a
1
+
1
+
a
2
a
2
+
1
+
⋯
+
a
22
a
22
+
1
<
16.
\frac{a_1}{a_1+1}+\frac{a_2}{a_2+1}+\dots+\frac{a_{22}}{a_{22}+1}<16.
a
1
+
1
a
1
+
a
2
+
1
a
2
+
⋯
+
a
22
+
1
a
22
<
16.
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