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National and Regional Contests
Poland Contests
Poland - Second Round
2012 Poland - Second Round
1
1
Part of
2012 Poland - Second Round
Problems
(2)
System of equations
Source: Polish MO second round 2012
2/19/2012
a
,
b
,
c
,
d
∈
R
a,b,c,d\in\mathbb{R}
a
,
b
,
c
,
d
∈
R
, solve the system of equations:
{
a
3
+
b
=
c
b
3
+
c
=
d
c
3
+
d
=
a
d
3
+
a
=
b
\begin{cases} a^3+b=c \\ b^3+c=d \\ c^3+d=a \\ d^3+a=b \end{cases}
⎩
⎨
⎧
a
3
+
b
=
c
b
3
+
c
=
d
c
3
+
d
=
a
d
3
+
a
=
b
algebra unsolved
algebra
Functional equation
Source: Polish MO second round 2012
2/19/2012
f
,
g
:
R
→
R
f,g:\mathbb{R}\rightarrow\mathbb{R}
f
,
g
:
R
→
R
find all
f
,
g
f,g
f
,
g
satisfying
∀
x
,
y
∈
R
\forall x,y\in \mathbb{R}
∀
x
,
y
∈
R
:
g
(
f
(
x
)
−
y
)
=
f
(
g
(
y
)
)
+
x
.
g(f(x)-y)=f(g(y))+x.
g
(
f
(
x
)
−
y
)
=
f
(
g
(
y
))
+
x
.
function
algebra
functional equation