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Contests
International Contests
Pan African
2005 Pan African
1
1
Part of
2005 Pan African
Problems
(2)
AM-HM like Inequality
Source: Pan African Maths Olympiad
8/11/2005
For any positive real numbers
a
,
b
a,b
a
,
b
and
c
c
c
, prove:
1
a
+
1
b
+
1
c
≥
2
a
+
b
+
2
b
+
c
+
2
c
+
a
≥
9
a
+
b
+
c
\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} \geq \dfrac{2}{a+b} + \dfrac{2}{b+c} + \dfrac{2}{c+a} \geq \dfrac{9}{a+b+c}
a
1
+
b
1
+
c
1
≥
a
+
b
2
+
b
+
c
2
+
c
+
a
2
≥
a
+
b
+
c
9
inequalities
Greatest Integer Equation
Source: Pan African Maths Olympiad
8/11/2005
Let
[
x
]
[ {x} ]
[
x
]
be the greatest integer less than or equal to
x
x
x
, and let
{
x
}
=
x
−
[
x
]
\{x\}=x-[x]
{
x
}
=
x
−
[
x
]
. Solve the equation:
[
x
]
⋅
{
x
}
=
2005
x
[x] \cdot \{x\} = 2005x
[
x
]
⋅
{
x
}
=
2005
x