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Austrian-Polish
1995 Austrian-Polish Competition
1
a_n = a_{n-1} + a_{n-2} , nxn system
a_n = a_{n-1} + a_{n-2} , nxn system
Source: Austrian - Polish 1995 APMC
May 3, 2020
algebra
system of equations
Problem Statement
Determine all real solutions
(
a
1
,
.
.
.
,
a
n
)
(a_1,...,a_n)
(
a
1
,
...
,
a
n
)
of the following system of equations:
{
a
3
=
a
2
+
a
1
a
4
=
a
3
+
a
2
.
.
.
a
n
=
a
n
−
1
+
a
n
−
2
a
1
=
a
n
+
a
n
−
1
a
2
=
a
1
+
a
n
\begin{cases}a_3 = a_2 + a_1\\ a_4 = a_3 + a_2\\ ...\\ a_n = a_{n-1} + a_{n-2}\\ a_1= a_n +a_{n-1} \\ a_2 = a_1 + a_n \end{cases}
⎩
⎨
⎧
a
3
=
a
2
+
a
1
a
4
=
a
3
+
a
2
...
a
n
=
a
n
−
1
+
a
n
−
2
a
1
=
a
n
+
a
n
−
1
a
2
=
a
1
+
a
n
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