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1990 All Soviet Union Mathematical Olympiad
520
ASU 520 All Soviet Union MO 1990 sum x_i^2/(x_i + x_{i+1}) >=1/2 if sum x_1=1
ASU 520 All Soviet Union MO 1990 sum x_i^2/(x_i + x_{i+1}) >=1/2 if sum x_1=1
Source:
August 14, 2019
algebra
inequalities
minimum
Problem Statement
Let
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2, ..., x_n
x
1
,
x
2
,
...
,
x
n
be positive reals with sum
1
1
1
. Show that
x
1
2
x
1
+
x
2
+
x
2
2
x
2
+
x
3
+
.
.
.
+
x
n
−
1
2
x
n
−
1
+
x
n
+
x
n
2
x
n
+
x
1
≥
1
2
\frac{x_1^2}{x_1 + x_2}+ \frac{x_2^2}{x_2 + x_3} +... + \frac{x_{n-1}^2}{x_{n-1} + x_n} + \frac{x_n^2}{x_n + x_1} \ge \frac12
x
1
+
x
2
x
1
2
+
x
2
+
x
3
x
2
2
+
...
+
x
n
−
1
+
x
n
x
n
−
1
2
+
x
n
+
x
1
x
n
2
≥
2
1
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