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min of a/(b+c-a)+4b/(c+a-b)+ 9c/(a+b-c), if a+b+c=12 (HOMC 2017 J Q13)
min of a/(b+c-a)+4b/(c+a-b)+ 9c/(a+b-c), if a+b+c=12 (HOMC 2017 J Q13)
Source:
August 7, 2019
algebra
minimum
inequalities
Problem Statement
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be the side-lengths of triangle
A
B
C
ABC
A
BC
with
a
+
b
+
c
=
12
a+b+c = 12
a
+
b
+
c
=
12
. Determine the smallest value of
M
=
a
b
+
c
−
a
+
4
b
c
+
a
−
b
+
9
c
a
+
b
−
c
M =\frac{a}{b + c - a}+\frac{4b}{c + a - b}+\frac{9c}{a + b - c}
M
=
b
+
c
−
a
a
+
c
+
a
−
b
4
b
+
a
+
b
−
c
9
c
.
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