![\[ (a^2+3)(b^2+3)(c^2+3)(d^2+3) + k(a-1)(b-1)(c-1)(d-1) \ge 0 \]](http://latex.artofproblemsolving.com/4/d/b/4db42ebabc7309bba9629a9a7125be43abf113f0.png)
holds for all real numbers 
,
,
.
such that 
.
at points 
respectively. Let 
and 
be points on the segment 
respectively such that 
is parallel to 
.
be the circle that is externally tangent with the excircle
.
,
,
is a bijection and 
is an integer such that 
Show that 
is a multiple of 
Y by
Let 
be a prime and 
a nonempty set. Suppose that for each element 
there exist elements 
, 
such that 
is divisible by . Prove that 
, where 
denotes the cardinality of 
.
be a prime and 
a nonempty set. Suppose that for each element 
there exist elements 
, 
such that 
is divisible by . Prove that 
, where 
denotes the cardinality of 
.
