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2020 BMT Fall
7
BMT Algebra #7 - Symmetric Equations
BMT Algebra #7 - Symmetric Equations
Source:
October 11, 2020
Bmt
algebra
Problem Statement
Let
a
,
b
,
a,\,b,
a
,
b
,
and
c
c
c
be real numbers such that
a
+
b
+
c
=
1
a
+
1
b
+
1
c
a+b+c=\frac1{a}+\frac1{b}+\frac1{c}
a
+
b
+
c
=
a
1
+
b
1
+
c
1
and
a
b
c
=
5
abc=5
ab
c
=
5
. The value of
(
a
−
1
b
)
3
+
(
b
−
1
c
)
3
+
(
c
−
1
a
)
3
\left(a-\frac1{b}\right)^3+\left(b-\frac1{c}\right)^3+\left(c-\frac1{a}\right)^3
(
a
−
b
1
)
3
+
(
b
−
c
1
)
3
+
(
c
−
a
1
)
3
can be written in the form
m
n
\tfrac{m}{n}
n
m
, where
m
m
m
and
n
n
n
are relatively prime positive integers. Compute
m
+
n
m+n
m
+
n
.
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