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2018-IMOC
A7
system 3=ab+bc+ca=de+ef+fd=..., prove cf+fi+ic=3
system 3=ab+bc+ca=de+ef+fd=..., prove cf+fi+ic=3
Source: IMOC 2018 A7
August 16, 2021
algebra
system of equations
Problem Statement
If the reals
a
,
b
,
c
,
d
,
e
,
f
,
g
,
h
,
i
a,b,c,d,e,f,g,h,i
a
,
b
,
c
,
d
,
e
,
f
,
g
,
h
,
i
satisfy
{
a
b
+
b
c
+
c
a
=
3
d
e
+
e
f
+
f
d
=
3
g
h
+
h
i
+
i
g
=
3
a
d
+
d
g
+
g
a
=
3
b
e
+
e
h
+
h
b
=
3
\begin{cases}ab+bc+ca=3\\de+ef+fd=3\\gh+hi+ig=3\\ad+dg+ga=3\\be+eh+hb=3\end{cases}
⎩
⎨
⎧
ab
+
b
c
+
c
a
=
3
d
e
+
e
f
+
fd
=
3
g
h
+
hi
+
i
g
=
3
a
d
+
d
g
+
g
a
=
3
b
e
+
e
h
+
hb
=
3
show that
c
f
+
f
i
+
i
c
=
3
cf+fi+ic=3
c
f
+
f
i
+
i
c
=
3
holds as well.
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