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National and Regional Contests
China Contests
China Northern MO
2009 China Northern MO
5
sum (x^{2009}-2008(x-1))/(y+z) >= 1/2 (x+y+z), if x,y,z>0, x^2+y^2+z^2 = 3
sum (x^{2009}-2008(x-1))/(y+z) >= 1/2 (x+y+z), if x,y,z>0, x^2+y^2+z^2 = 3
Source: China Northern MO 2009 p5 CNMO
December 12, 2020
algebra
inequalities
Problem Statement
Assume :
x
,
y
,
z
>
0
x,y,z>0
x
,
y
,
z
>
0
,
x
2
+
y
2
+
z
2
=
3
x^2+y^2+z^2 = 3
x
2
+
y
2
+
z
2
=
3
. Prove the following inequality :
x
2009
−
2008
(
x
−
1
)
y
+
z
+
y
2009
−
2008
(
y
−
1
)
x
+
z
+
z
2009
−
2008
(
z
−
1
)
x
+
y
≥
1
2
(
x
+
y
+
z
)
{\frac{x^{2009}-2008(x-1)}{y+z}+\frac{y^{2009}-2008(y-1)}{x+z}+\frac{z^{2009}-2008(z-1)}{x+y}\ge\frac{1}{2}(x+y+z)}
y
+
z
x
2009
−
2008
(
x
−
1
)
+
x
+
z
y
2009
−
2008
(
y
−
1
)
+
x
+
y
z
2009
−
2008
(
z
−
1
)
≥
2
1
(
x
+
y
+
z
)
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