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Taiwan APMO Prelininary
2020 Taiwan APMO Preliminary
P6
2020 Taiwan APMO Preliminary Problem 6
2020 Taiwan APMO Preliminary Problem 6
Source: 2020 Taiwan APMO Preliminary
July 23, 2020
algebra
inequalities
Inequality
Problem Statement
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive reals. Find the minimum value of
13
a
+
13
b
+
2
c
2
a
+
2
b
+
24
a
−
b
+
13
c
2
b
+
2
c
+
(
−
a
+
24
b
+
13
c
)
2
c
+
2
a
\dfrac{13a+13b+2c}{2a+2b}+\dfrac{24a-b+13c}{2b+2c}+\dfrac{(-a+24b+13c)}{2c+2a}
2
a
+
2
b
13
a
+
13
b
+
2
c
+
2
b
+
2
c
24
a
−
b
+
13
c
+
2
c
+
2
a
(
−
a
+
24
b
+
13
c
)
. (1)What is the minimum value? (2)If the minimum value occurs when
(
a
,
b
,
c
)
=
(
a
0
,
b
0
,
c
0
)
(a,b,c)=(a_0,b_0,c_0)
(
a
,
b
,
c
)
=
(
a
0
,
b
0
,
c
0
)
,then find
b
0
a
0
+
c
0
b
0
\frac{b_0}{a_0}+\frac{c_0}{b_0}
a
0
b
0
+
b
0
c
0
.
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