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Canada Contests
Canadian Mathematical Olympiad Qualification Repechage
2010 Canadian Mathematical Olympiad Qualification Repechage
7
Function g for triples of real numbers
Function g for triples of real numbers
Source: Canadian Repêchage 2010: Problem 7
May 6, 2014
function
algebra solved
algebra
Problem Statement
If
(
a
,
b
,
c
)
(a,~b,~c)
(
a
,
b
,
c
)
is a triple of real numbers, de fine [*]
g
(
a
,
b
,
c
)
=
(
a
+
b
,
b
+
c
,
a
+
c
)
g(a,~b,~c)=(a+b,~b+c,~a+c)
g
(
a
,
b
,
c
)
=
(
a
+
b
,
b
+
c
,
a
+
c
)
, and [*]
g
n
(
a
,
b
,
c
)
=
g
(
g
n
−
1
(
a
,
b
,
c
)
)
g^n(a,~b,~c)=g(g^{n-1}(a,~b,~c))
g
n
(
a
,
b
,
c
)
=
g
(
g
n
−
1
(
a
,
b
,
c
))
for
n
≥
2
n\ge 2
n
≥
2
Suppose that there exists a positive integer
n
n
n
so that
g
n
(
a
,
b
,
c
)
=
(
a
,
b
,
c
)
g^n(a,~b,~c)=(a,~b,~c)
g
n
(
a
,
b
,
c
)
=
(
a
,
b
,
c
)
for some
(
a
,
b
,
c
)
≠
(
0
,
0
,
0
)
(a,~b,~c)\neq (0,~0,~0)
(
a
,
b
,
c
)
=
(
0
,
0
,
0
)
. Prove that
g
6
(
a
,
b
,
c
)
=
(
a
,
b
,
c
)
g^6(a,~b,~c)=(a,~b,~c)
g
6
(
a
,
b
,
c
)
=
(
a
,
b
,
c
)
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