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Austria Contests
Austrian MO National Competition
2005 Federal Competition For Advanced Students, Part 1
3
3
Part of
2005 Federal Competition For Advanced Students, Part 1
Problems
(1)
$a^n+b^n+c^n$
Source: Austrian MO 2005 round 1
6/27/2005
For 3 real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
let
s
n
=
a
n
+
b
n
+
c
n
s_n=a^{n}+b^{n}+c^{n}
s
n
=
a
n
+
b
n
+
c
n
. It is known that
s
1
=
2
s_1=2
s
1
=
2
,
s
2
=
6
s_2=6
s
2
=
6
and
s
3
=
14
s_3=14
s
3
=
14
. Prove that for all natural numbers
n
>
1
n>1
n
>
1
, we have
∣
s
n
2
−
s
n
−
1
s
n
+
1
∣
=
8
|s^2_n-s_{n-1}s_{n+1}|=8
∣
s
n
2
−
s
n
−
1
s
n
+
1
∣
=
8
algebra unsolved
algebra