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Moldova National Olympiad
2006 Moldova National Olympiad
10.5
Moldova mo 2006, 10th grade
Moldova mo 2006, 10th grade
Source: Moldova MO 2006
March 21, 2006
logarithms
inequalities
calculus
derivative
function
algebra unsolved
algebra
Problem Statement
Let
x
1
x_{1}
x
1
,
x
2
x_{2}
x
2
,
…
\ldots
…
,
x
n
x_{n}
x
n
be
n
n
n
real numbers in
(
1
4
,
2
3
)
\left(\frac{1}{4},\frac{2}{3}\right)
(
4
1
,
3
2
)
. Find the minimal value of the expression:
log
3
2
x
1
(
1
2
−
1
36
x
2
2
)
+
log
3
2
x
2
(
1
2
−
1
36
x
3
2
)
+
⋯
+
log
3
2
x
n
(
1
2
−
1
36
x
1
2
)
.
\log_{\frac 32x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\log_{\frac 32x_{2}}\left(\frac{1}{2}-\frac{1}{36x_{3}^{2}}\right)+\cdots+ \log_{\frac 32x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right).
lo
g
2
3
x
1
(
2
1
−
36
x
2
2
1
)
+
lo
g
2
3
x
2
(
2
1
−
36
x
3
2
1
)
+
⋯
+
lo
g
2
3
x
n
(
2
1
−
36
x
1
2
1
)
.
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