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All-Russian Olympiad Regional Round
2000 All-Russian Olympiad Regional Round
11.3
11.3
Part of
2000 All-Russian Olympiad Regional Round
Problems
(1)
sum a_i^3 = (sum a_i)^2 - All-Russian MO 2000 Regional (R4) 11.3
Source:
9/26/2024
Sequence of real numbers
a
1
,
a
2
,
.
.
.
,
a
2000
a_1, a_2, . . . , a_{2000}
a
1
,
a
2
,
...
,
a
2000
is such that for any natural number
n
n
n
,
1
≤
n
≤
2000
1\le n \le 2000
1
≤
n
≤
2000
, the equality
a
1
3
+
a
2
3
+
.
.
.
+
a
n
3
=
(
a
1
+
a
2
+
.
.
.
+
a
n
)
2
.
a^3_1+ a^3_2+... + a^3_n = (a_1 + a_2 +...+ a_n)^2.
a
1
3
+
a
2
3
+
...
+
a
n
3
=
(
a
1
+
a
2
+
...
+
a
n
)
2
.
Prove that all terms of this sequence are integers.
algebra
number theory
Integer