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National and Regional Contests
Singapore Contests
Singapore Senior Math Olympiad
2011 Singapore Senior Math Olympiad
5
5
Part of
2011 Singapore Senior Math Olympiad
Problems
(1)
Inequality
Source: Singapore MO 2011 senior round 2 Q5
6/25/2011
Given
x
1
,
x
2
,
…
,
x
n
>
0
,
n
≥
5
x_1,x_2,\dots,x_n>0,n\geq 5
x
1
,
x
2
,
…
,
x
n
>
0
,
n
≥
5
, show that
x
1
x
2
x
1
2
+
x
2
2
+
2
x
3
x
4
+
x
2
x
3
x
2
2
+
x
3
2
+
2
x
4
x
5
+
⋯
+
x
n
x
1
x
n
2
+
x
1
2
+
2
x
2
x
3
≤
n
−
1
2
\frac{x_1x_2}{x_1^2+x_2^2+2x_3x_4}+\frac{x_2x_3}{x_2^2+x_3^2+2x_4x_5}+\cdots+\frac{x_nx_1}{x_n^2+x_1^2+2x_2x_3}\leq \frac{n-1}{2}
x
1
2
+
x
2
2
+
2
x
3
x
4
x
1
x
2
+
x
2
2
+
x
3
2
+
2
x
4
x
5
x
2
x
3
+
⋯
+
x
n
2
+
x
1
2
+
2
x
2
x
3
x
n
x
1
≤
2
n
−
1
inequalities
inequalities proposed