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All-Russian Olympiad
1964 All Russian Mathematical Olympiad
044
044
Part of
1964 All Russian Mathematical Olympiad
Problems
(1)
ASU 044 All Russian MO 1964 8.4 sets of halfs of integers
Source:
6/18/2019
Given an arbitrary set of
2
k
+
1
2k+1
2
k
+
1
integers
{
a
1
,
a
2
,
.
.
.
,
a
2
k
+
1
}
\{a_1,a_2,...,a_{2k+1}\}
{
a
1
,
a
2
,
...
,
a
2
k
+
1
}
. We make a new set
{
(
a
1
+
a
2
)
/
2
,
(
a
2
+
a
3
)
/
2
,
(
a
2
k
+
a
2
k
+
1
)
/
2
,
(
a
2
k
+
1
+
a
1
)
/
2
}
\{(a_1+a_2)/2, (a_2+a_3)/2, (a_{2k}+a_{2k+1})/2, (a_{2k+1}+a_1)/2\}
{(
a
1
+
a
2
)
/2
,
(
a
2
+
a
3
)
/2
,
(
a
2
k
+
a
2
k
+
1
)
/2
,
(
a
2
k
+
1
+
a
1
)
/2
}
and a new one, according to the same rule, and so on... Prove that if we obtain integers only, the initial set consisted of equal integers only.
number theory