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1985 IMO Shortlist
18
n variables ineq. IMO Long List 1986
n variables ineq. IMO Long List 1986
Source:
August 29, 2010
n-variable inequality
Inequality
algebra
IMO Shortlist
Problem Statement
Let
x
1
,
x
2
,
⋯
,
x
n
x_1, x_2, \cdots , x_n
x
1
,
x
2
,
⋯
,
x
n
be positive numbers. Prove that
x
1
2
x
1
2
+
x
2
x
3
+
x
2
2
x
2
2
+
x
3
x
4
+
⋯
+
x
n
−
1
2
x
n
−
1
2
+
x
n
x
1
+
x
n
2
x
n
2
+
x
1
x
2
≤
n
−
1
\frac{x_1^2}{x_1^2+x_2x_3} + \frac{x_2^2}{x_2^2+x_3x_4} + \cdots +\frac{x_{n-1}^2}{x_{n-1}^2+x_nx_1} +\frac{x_n^2}{x_n^2+x_1x_2} \leq n-1
x
1
2
+
x
2
x
3
x
1
2
+
x
2
2
+
x
3
x
4
x
2
2
+
⋯
+
x
n
−
1
2
+
x
n
x
1
x
n
−
1
2
+
x
n
2
+
x
1
x
2
x
n
2
≤
n
−
1
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