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May Olympiad
2021 May Olympiad
5
100 numbers and 1 perfect cube
100 numbers and 1 perfect cube
Source: May Olympiad L2 #5
June 8, 2021
number theory
Problem Statement
Prove that there are
100
100
100
distinct positive integers
n
1
,
n
2
,
…
,
n
99
,
n
100
n_1,n_2,\dots,n_{99},n_{100}
n
1
,
n
2
,
…
,
n
99
,
n
100
such that
n
1
3
+
n
2
3
+
⋯
+
n
100
3
100
\frac{n_1^3+n_2 ^3+\dots +n_{100}^3}{100}
100
n
1
3
+
n
2
3
+
⋯
+
n
100
3
is a perfect cube.
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