MathDB
Problems
Contests
National and Regional Contests
Moldova Contests
JBMO TST - Moldova
2002 Junior Balkan Team Selection Tests - Moldova
12
12
Part of
2002 Junior Balkan Team Selection Tests - Moldova
Problems
(1)
f (-x) = -f (x) if f (g (x)) = g (f (x)) = x, f (x) + g (x) = x
Source: 2002 Moldova JBMO TST p12
2/25/2021
Let
M
M
M
be an empty set of real numbers. For any
x
∈
M
x \in M
x
∈
M
the functions
f
:
M
→
M
f: M\to M
f
:
M
→
M
and
g
:
M
→
M
g: M\to M
g
:
M
→
M
satisfy the relations
f
(
g
(
x
)
)
=
g
(
f
(
x
)
)
=
x
f (g (x)) = g (f (x)) = x
f
(
g
(
x
))
=
g
(
f
(
x
))
=
x
and
f
(
x
)
+
g
(
x
)
=
x
f (x) + g (x) = x
f
(
x
)
+
g
(
x
)
=
x
. Show that
−
x
∈
M
- x \in M
−
x
∈
M
¸ and
f
(
−
x
)
=
−
f
(
x
)
f (-x) = -f (x)
f
(
−
x
)
=
−
f
(
x
)
whatever
x
∈
M
x \in M
x
∈
M
.
algebra
functions