,![\[
\sqrt{a + b} + \sqrt{b + c} + \sqrt{c + a} \geq \frac{4(ab + bc + ca)}{\sqrt{(a + b)(b + c)(c + a)}}
\]](http://latex.artofproblemsolving.com/e/7/2/e72fd0f134289c359bc8766440babc52ae131961.png)
satisfied 
Find the minimum of 
satisfy that: 
and natural number 
,
are there such that 
and 
?
and 
.
be real numbers such that 
Prove that
Let 
Prove that
and 
, then we can deduce 
\le 2 $ ?
satisfying 
.![\[
0 \le (x^2 + 1)(y^2 + 1) + 4(x - 1)(y - 1) \le 164.
\]](http://latex.artofproblemsolving.com/4/2/d/42d0b88388462f16dd99d928b94024e689ba7c2e.png)
Y by
Let 
be a trapezoid with 
, such that 
. Points 
and 
divide 
into three equal parts, is between 
and . Lines 
and 
intersect 
. Show that 
.
be a trapezoid with 
, such that 
. Points 
and 
divide 
into three equal parts, is between 
and . Lines 
and 
intersect 
. Show that 
.
