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be a positive integer. Define 
as the number of integer pairs 
satisfying 
with 
odd. Define 
as the number of integer pairs satisfying 
of 
for which 
divides the number ![\[ 1990^{1991^{1992}} + 1992^{1991^{1990}}.\]](http://latex.artofproblemsolving.com/c/0/6/c06d8be837b5996a8c7600460962ce0a74e3b471.png)
?
such that 
is a perfect cube.
digits are distinct?
Solve for .
, 
, 
, and 
. Let 
be the point on 
such that 
, and let 
be the midpoint of 
. If 
is a point such that 
is a rectangle, compute the area of 
.
has a sidelength of . Points 
, 
, 
, and 
are chosen on 
, , 
, and respectively, such that 
, 
, 
, and 
are length 
. Compute the area of quadrilateral 
.
satisfies for all integers , 
If 
and 
, compute 
.
children have black hair, all but 
have red hair, and all but 
have blond hair. How many children are there in the class? such that the set 
can be partitioned into n Akash subsets. be a natural number and be a prime number. How many different pairs 
satisfy the equation:
, ![$ x \in [-1 / 10,0] $](http://latex.artofproblemsolving.com/1/e/f/1ef43e8c80ca68c6262a0f96555efb0a00e531d1.png)
. Calculate the function where 
is the inverse function of 
]
different coloured marbles on it and another one is empty
be an integer and denote 
. For a function 
we denote Im 
, Fix 
and 
. Find all possible values of 
Im 
+ Fix + 
.
and 
Show that
Let 
Show that
:
.
Let 
.
