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National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1963 Dutch Mathematical Olympiad
4
4
Part of
1963 Dutch Mathematical Olympiad
Problems
(1)
a_3+a_3+...+a_{2n-2}=0 if (x^2-x+1)^n - (x^2-x+2)^n+ (1+x)^n+(2-x)^n
Source: Netherlands - Dutch NMO 1963 p4
1/31/2023
One considers for
n
>
2
n > 2
n
>
2
the polynomial:
(
x
2
−
x
+
1
)
n
−
(
x
2
−
x
+
2
)
n
+
(
1
+
x
)
n
+
(
2
−
x
)
n
(x^2-x+1)^n - (x^2-x+2)^n+ (1+x)^n+(2-x)^n
(
x
2
−
x
+
1
)
n
−
(
x
2
−
x
+
2
)
n
+
(
1
+
x
)
n
+
(
2
−
x
)
n
Show that the degree of this polynomial is
2
n
−
2
2n - 2
2
n
−
2
. The polynomial is written in the form
a
0
+
a
1
x
+
a
2
x
2
+
.
.
.
+
a
2
n
−
2
x
2
n
−
2
a_0+a_1x+a_2x^2+...+a_{2n-2}x^{2n-2}
a
0
+
a
1
x
+
a
2
x
2
+
...
+
a
2
n
−
2
x
2
n
−
2
Prove that
a
2
+
a
3
+
.
.
.
+
a
2
n
−
2
=
0
a_2+a_3+...+a_{2n-2}=0
a
2
+
a
3
+
...
+
a
2
n
−
2
=
0
polynomial
algebra