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Balkan MO
1990 Balkan MO
2
a polynomial equality
a polynomial equality
Source: bmo 1990
April 23, 2007
algebra
polynomial
algebra proposed
Problem Statement
The polynomial
P
(
X
)
P(X)
P
(
X
)
is defined by
P
(
X
)
=
(
X
+
2
X
2
+
…
+
n
X
n
)
2
=
a
0
+
a
1
X
+
…
+
a
2
n
X
2
n
P(X)=(X+2X^{2}+\ldots +nX^{n})^{2}=a_{0}+a_{1}X+\ldots +a_{2n}X^{2n}
P
(
X
)
=
(
X
+
2
X
2
+
…
+
n
X
n
)
2
=
a
0
+
a
1
X
+
…
+
a
2
n
X
2
n
. Prove that
a
n
+
1
+
a
n
+
2
+
…
+
a
2
n
=
n
(
n
+
1
)
(
5
n
2
+
5
n
+
2
)
24
a_{n+1}+a_{n+2}+\ldots +a_{2n}=\frac{n(n+1)(5n^{2}+5n+2)}{24}
a
n
+
1
+
a
n
+
2
+
…
+
a
2
n
=
24
n
(
n
+
1
)
(
5
n
2
+
5
n
+
2
)
.
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