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Fraction is greater than 1 - JBMO Shortlist
Fraction is greater than 1 - JBMO Shortlist
Source:
October 30, 2010
algebra proposed
algebra
Problem Statement
Let
x
,
y
,
a
,
b
x,y,a,b
x
,
y
,
a
,
b
be positive real numbers such that
x
≠
y
x\not= y
x
=
y
,
x
≠
2
y
x\not= 2y
x
=
2
y
,
y
≠
2
x
y\not= 2x
y
=
2
x
,
a
≠
3
b
a\not=3b
a
=
3
b
and
2
x
−
y
2
y
−
x
=
a
+
3
b
a
−
3
b
\frac{2x-y}{2y-x}=\frac{a+3b}{a-3b}
2
y
−
x
2
x
−
y
=
a
−
3
b
a
+
3
b
. Prove that
x
2
+
y
2
x
2
−
y
2
≥
1
\frac{x^2+y^2}{x^2-y^2}\ge 1
x
2
−
y
2
x
2
+
y
2
≥
1
.
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