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Austrian-Polish
1999 Austrian-Polish Competition
3
3
Part of
1999 Austrian-Polish Competition
Problems
(1)
f_n(x^n)-f_1 (x)f_1(y)+ f_n(y^n) = 0, nxn functional system
Source: Austrian - Polish 1999 APMC
5/4/2020
Given an integer
n
≥
2
n \ge 2
n
≥
2
, find all sustems of
n
n
n
functions
f
1
,
.
.
.
,
f
n
:
R
→
R
f_1,..., f_n : R \to R
f
1
,
...
,
f
n
:
R
→
R
such that for all
x
,
y
∈
R
x,y \in R
x
,
y
∈
R
{
f
1
(
x
)
−
f
2
(
x
)
f
2
(
y
)
+
f
1
(
y
)
=
0
f
2
(
x
2
)
−
f
3
(
x
)
f
3
(
y
)
+
f
2
(
y
2
)
=
0
.
.
.
f
n
(
x
n
)
−
f
1
(
x
)
f
1
(
y
)
+
f
n
(
y
n
)
=
0
\begin{cases} f_1(x)-f_2 (x)f_2(y)+ f_1(y) = 0 \\ f_2(x^2)-f_3 (x)f_3(y)+ f_2(y^2) = 0 \\ ... \\ f_n(x^n)-f_1 (x)f_1(y)+ f_n(y^n) = 0 \end {cases}
⎩
⎨
⎧
f
1
(
x
)
−
f
2
(
x
)
f
2
(
y
)
+
f
1
(
y
)
=
0
f
2
(
x
2
)
−
f
3
(
x
)
f
3
(
y
)
+
f
2
(
y
2
)
=
0
...
f
n
(
x
n
)
−
f
1
(
x
)
f
1
(
y
)
+
f
n
(
y
n
)
=
0
system of equations
functional equation
functional
algebra