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National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2009 Korea Junior Math Olympiad
3
3
Part of
2009 Korea Junior Math Olympiad
Problems
(1)
\frac{x^2}{x+y}+\frac{y^2}{1-x}+\frac{(1-x-y)^2}{1-y}\geq\frac{1}{2} if 0<x,y<1
Source: KJMO 2009 p3
5/2/2019
For two arbitrary reals
x
,
y
x, y
x
,
y
which are larger than
0
0
0
and less than
1.
1.
1.
Prove that
x
2
x
+
y
+
y
2
1
−
x
+
(
1
−
x
−
y
)
2
1
−
y
≥
1
2
.
\frac{x^2}{x+y}+\frac{y^2}{1-x}+\frac{(1-x-y)^2}{1-y}\geq\frac{1}{2}.
x
+
y
x
2
+
1
−
x
y
2
+
1
−
y
(
1
−
x
−
y
)
2
≥
2
1
.
algebra
inequalities
Inequality