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2021-IMOC
A2
A2
Part of
2021-IMOC
Problems
(1)
IMOC 2021 A2
Source: IMOC 2021 A2
8/12/2021
For any positive integers
n
n
n
, find all
n
n
n
-tuples of complex numbers
(
a
1
,
.
.
.
,
a
n
)
(a_1,..., a_n)
(
a
1
,
...
,
a
n
)
satisfying
(
x
+
a
1
)
(
x
+
a
2
)
⋯
(
x
+
a
n
)
=
x
n
+
(
n
1
)
a
1
x
n
−
1
+
(
n
2
)
a
2
2
x
n
−
2
+
⋯
+
(
n
n
−
1
)
a
n
−
1
n
−
1
+
(
n
n
)
a
n
n
.
(x+a_1)(x+a_2)\cdots (x+a_n)=x^n+\binom{n}{1}a_1 x^{n-1}+\binom{n}{2}a_2^2 x^{n-2}+\cdots +\binom{n}{n-1} a_{n-1}^{n-1}+\binom{n}{n}a_n^n.
(
x
+
a
1
)
(
x
+
a
2
)
⋯
(
x
+
a
n
)
=
x
n
+
(
1
n
)
a
1
x
n
−
1
+
(
2
n
)
a
2
2
x
n
−
2
+
⋯
+
(
n
−
1
n
)
a
n
−
1
n
−
1
+
(
n
n
)
a
n
n
.
Proposed by USJL.
algebra
IMOC