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International Contests
Balkan MO Shortlist
2007 Balkan MO Shortlist
A5
function hard
function hard
Source:
June 20, 2017
function
algebra
Problem Statement
find all the function
f
,
g
:
R
→
R
f,g:R\rightarrow R
f
,
g
:
R
→
R
such that (1)for every
x
,
y
∈
R
x,y\in R
x
,
y
∈
R
we have
f
(
x
g
(
y
+
1
)
)
+
y
=
x
f
(
y
)
+
f
(
x
+
g
(
y
)
)
f(xg(y+1))+y=xf(y)+f(x+g(y))
f
(
xg
(
y
+
1
))
+
y
=
x
f
(
y
)
+
f
(
x
+
g
(
y
))
(2)
f
(
0
)
+
g
(
0
)
=
0
f(0)+g(0)=0
f
(
0
)
+
g
(
0
)
=
0
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