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Israel National Olympiad
2022 Israel National Olympiad
P2
Symbolic algebraic manipulation
Symbolic algebraic manipulation
Source: 2022 Israel National Olympiad P2
December 16, 2022
algebra
national olympiad
Problem Statement
Real nonzero numbers
a
,
b
,
c
,
d
,
e
,
f
,
k
,
m
a,b,c,d,e,f,k,m
a
,
b
,
c
,
d
,
e
,
f
,
k
,
m
satisfy the equations
a
b
+
c
d
+
e
f
=
k
\frac{a}{b}+\frac{c}{d}+\frac{e}{f}=k
b
a
+
d
c
+
f
e
=
k
b
c
+
d
e
+
f
a
=
m
\frac{b}{c}+\frac{d}{e}+\frac{f}{a}=m
c
b
+
e
d
+
a
f
=
m
a
d
=
b
e
=
c
f
ad=be=cf
a
d
=
b
e
=
c
f
Express
a
c
+
c
e
+
e
a
+
b
d
+
d
f
+
f
b
\frac{a}{c}+\frac{c}{e}+\frac{e}{a}+\frac{b}{d}+\frac{d}{f}+\frac{f}{b}
c
a
+
e
c
+
a
e
+
d
b
+
f
d
+
b
f
using
m
m
m
and
k
k
k
.
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