such that:
be a natural number and 
be a prime number. How many different pairs 
satisfy the equation:
such that 
and 
are both perfect cubes.
and 
Show that
Let 
Show that
The altitudes 
, intersect
.
and 
o
.
b
.
intersects 
,
intersects 
at 
lie on the same circle.
intersects the circle 
The line 
.
are collinear.
such that 
is an exact 5th power, 
is an exact 6th power, and 
is an exact 7th power.
denote the set of natural numbers and let 
denote 
ordered pairs in 
.
distinct elements in the set 
for 
whose product is a perfect cube.
. Prove that
have real roots and real coefficients with 
.
?
,
.
have roots 
.
.
(I won't be proving this here). This new information allows us to utilize the mean inequality chain as follows:
Similarly, applying AM-HM yields 
.
.
such that 
is a perfect cube.
digits are distinct?
Solve for 
,
,
,
.
be the point on 
,
be the midpoint of 
.
is a point such that 
is a rectangle, compute the area of 
.
has a sidelength of 
,
,
,
,
,
,
,
are length 
.
.
satisfies for all integers 
If 
and 
,
.
children have black hair, all but 
have red hair, and all but 
can be partitioned into n Akash subsets.
:
.
Let 
.
Y by
Suppose 
die is laid down on an infinite grid of 
squares. The die starts perfectly aligned in one of the square of the grid with the 
facing up and the 
facing forward. The die can be "rolled" along an edge such that is moves exactly one unit up, down, left, or right and rotates 
correspondingly in that axis.
Determine, with proof, all possible squares relative to the starting square for which it is possible to roll the die to that square and have it in the same orientation ( facing up and facing forward).
(This problem may be hard to visualize/understand. If you don't understand the question, please ask and I can clarify.)
die is laid down on an infinite grid of 
squares. The die starts perfectly aligned in one of the square of the grid with the 
facing up and the 
facing forward. The die can be "rolled" along an edge such that is moves exactly one unit up, down, left, or right and rotates 
correspondingly in that axis.Determine, with proof, all possible squares relative to the starting square for which it is possible to roll the die to that square and have it in the same orientation (
facing up and facing forward).(This problem may be hard to visualize/understand. If you don't understand the question, please ask and I can clarify.)
