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Romania Contests
Romania Team Selection Test
1991 Romania Team Selection Test
7
7
Part of
1991 Romania Team Selection Test
Problems
(1)
x_1^2x_2^2...x_n^2+x_2^2x_3^2...x_{n+1}^2+...+x_{2n}^2x_1^2...x_{n-1}^2<1/n^{2n}
Source: Romania IMO TST 1991 p8
2/19/2020
Let
x
1
,
x
2
,
.
.
.
,
x
2
n
x_1,x_2,...,x_{2n}
x
1
,
x
2
,
...
,
x
2
n
be positive real numbers with the sum
1
1
1
. Prove that
x
1
2
x
2
2
.
.
.
x
n
2
+
x
2
2
x
3
2
.
.
.
x
n
+
1
2
+
.
.
.
+
x
2
n
2
x
1
2
.
.
.
x
n
−
1
2
<
1
n
2
n
x_1^2x_2^2...x_n^2+x_2^2x_3^2...x_{n+1}^2+...+x_{2n}^2x_1^2...x_{n-1}^2 <\frac{1}{n^{2n}}
x
1
2
x
2
2
...
x
n
2
+
x
2
2
x
3
2
...
x
n
+
1
2
+
...
+
x
2
n
2
x
1
2
...
x
n
−
1
2
<
n
2
n
1
inequalities
Sum
algebra