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Canada National Olympiad
1969 Canada National Olympiad
1
Odd equation (with equality condition of cauchy)
Odd equation (with equality condition of cauchy)
Source: Canada 1969, Problem 1
May 14, 2006
algebra unsolved
algebra
Problem Statement
If
a
1
/
b
1
=
a
2
/
b
2
=
a
3
/
b
3
a_1/b_1=a_2/b_2=a_3/b_3
a
1
/
b
1
=
a
2
/
b
2
=
a
3
/
b
3
and
p
1
,
p
2
,
p
3
p_1,p_2,p_3
p
1
,
p
2
,
p
3
are not all zero, show that for all
n
∈
N
n\in\mathbb{N}
n
∈
N
,
(
a
1
b
1
)
n
=
p
1
a
1
n
+
p
2
a
2
n
+
p
3
a
3
n
p
1
b
1
n
+
p
2
b
2
n
+
p
3
b
3
n
.
\left(\frac{a_1}{b_1}\right)^n = \frac{p_1a_1^n+p_2a_2^n+p_3a_3^n}{p_1b_1^n+p_2b_2^n+p_3b_3^n}.
(
b
1
a
1
)
n
=
p
1
b
1
n
+
p
2
b
2
n
+
p
3
b
3
n
p
1
a
1
n
+
p
2
a
2
n
+
p
3
a
3
n
.
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