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National and Regional Contests
India Contests
India National Olympiad
2006 India National Olympiad
3
3
Part of
2006 India National Olympiad
Problems
(1)
f(a,b,c) = (a+b+c, ab+bc+ca, abc)
Source: Indian National Maths Olympiad, Problem 3
2/5/2006
Let
X
=
Z
3
X=\mathbb{Z}^3
X
=
Z
3
denote the set of all triples
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
of integers. Define
f
:
X
ā
X
f: X \to X
f
:
X
ā
X
by
f
(
a
,
b
,
c
)
=
(
a
+
b
+
c
,
a
b
+
b
c
+
c
a
,
a
b
c
)
.
f(a,b,c) = (a+b+c, ab+bc+ca, abc) .
f
(
a
,
b
,
c
)
=
(
a
+
b
+
c
,
ab
+
b
c
+
c
a
,
ab
c
)
.
Find all triples
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
such that
f
(
f
(
a
,
b
,
c
)
)
=
(
a
,
b
,
c
)
.
f(f(a,b,c)) = (a,b,c) .
f
(
f
(
a
,
b
,
c
))
=
(
a
,
b
,
c
)
.
analytic geometry
algebra proposed
algebra