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Danube Competition in Mathematics
2019 Danube Mathematical Competition
2
n irational numbers
n irational numbers
Source: 2019 Danube
October 29, 2019
number theory
Problem Statement
Let be a natural number
n
,
n,
n
,
and
n
n
n
real numbers
a
1
,
a
2
,
…
,
a
n
.
a_1,a_2,\ldots ,a_n.
a
1
,
a
2
,
…
,
a
n
.
Prove that there exists a real number
a
a
a
such that
a
+
a
1
,
a
+
a
2
,
…
,
a
+
a
n
a+a_1,a+a_2,\ldots ,a+a_n
a
+
a
1
,
a
+
a
2
,
…
,
a
+
a
n
are all irrational.
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