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Korea Contests
Korea Junior Mathematics Olympiad
2014 Korea Junior Math Olympiad
2
s/x_1+t/x_2+s/x_3+t/x_4+...+s/x_{2n-1}+t/x_{2n} > 2n^2/(n+1)
s/x_1+t/x_2+s/x_3+t/x_4+...+s/x_{2n-1}+t/x_{2n} > 2n^2/(n+1)
Source: KJMO 2014 p2
May 2, 2019
algebra
Sum
inequalities
Inequality
Problem Statement
Let there be
2
n
2n
2
n
positive reals
a
1
,
a
2
,
.
.
.
,
a
2
n
a_1,a_2,...,a_{2n}
a
1
,
a
2
,
...
,
a
2
n
. Let
s
=
a
1
+
a
3
+
.
.
.
+
a
2
n
−
1
s = a_1 + a_3 +...+ a_{2n-1}
s
=
a
1
+
a
3
+
...
+
a
2
n
−
1
,
t
=
a
2
+
a
4
+
.
.
.
+
a
2
n
t = a_2 + a_4 + ... + a_{2n}
t
=
a
2
+
a
4
+
...
+
a
2
n
, and
x
k
=
a
k
+
a
k
+
1
+
.
.
.
+
a
k
+
n
−
1
x_k = a_k + a_{k+1} + ... + a_{k+n-1}
x
k
=
a
k
+
a
k
+
1
+
...
+
a
k
+
n
−
1
(indices are taken modulo
2
n
2n
2
n
). Prove that
s
x
1
+
t
x
2
+
s
x
3
+
t
x
4
+
.
.
.
+
s
x
2
n
−
1
+
t
x
2
n
>
2
n
2
n
+
1
\frac{s}{x_1}+\frac{t}{x_2}+\frac{s}{x_3}+\frac{t}{x_4}+...+\frac{s}{x_{2n-1}}+\frac{t}{x_{2n}}>\frac{2n^2}{n+1}
x
1
s
+
x
2
t
+
x
3
s
+
x
4
t
+
...
+
x
2
n
−
1
s
+
x
2
n
t
>
n
+
1
2
n
2
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