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be a convex quadrilateral and points 
and 
on sides 
such that![\[\tfrac{AB}{AE}=\tfrac{CD}{DF}=n\]](http://latex.artofproblemsolving.com/b/4/2/b42cd6305fab848a60c79689f17bdb55e1ba0d31.png)
is the area of 
show that 
and can teleport to other Cartesian coordinates with only 
of 
moves: 
, 
when 
, 
, and 
when 
.
be any polynomial with positive integer coefficients that passes through 
. Show that for all such , there exists a unique point on the curve where the wizard can land on., let be this unique point. Find the equation of the graph that contains all potential .
, let 
denote the increasing arithmetic sequence of integers whose first term is and whose common difference is . For example, 
is the sequence 
. For how many values of does contain the term 
?
is the set of positive real solutions to the system![\[
x^3 + y^3 + z^3 = x + y + z
\]](http://latex.artofproblemsolving.com/7/7/a/77a748a8b55a6b44761bced9bb1031193b6127dc.png)
![\[
x^2 + y^2 + z^2 = xyz
\]](http://latex.artofproblemsolving.com/9/3/a/93a4dffc6fd17b9f19f738d3b3865f56aa0151a9.png)
then find the number of elements in .
satisfy![\[
a + \frac{1}{a + 2015} = b - 4030 + \frac{1}{b - 2015}
\]](http://latex.artofproblemsolving.com/8/c/0/8c0b0cfdd569ea250e97590a3364794920d86e04.png)
and 
. Determine the value of![\[
\frac{ab}{2015} - a + b.
\]](http://latex.artofproblemsolving.com/1/f/6/1f620dea7a0c769137bf13004916e8b283e00eee.png)
, above the xy-plane, such that the pyramid 
is a regular tetrahedron.
with respect to a circle 
, a diameter 
, and a given point 
, such that is the reflection of the -antipode about ? Equivalently, is the the other intersection of and the line through parallel to .
are of the form 
, where 
, 2, 3, . For each 
, let 
be the greatest common divisor of 
and 
. Find the maximum value of as ranges through the positive integers.
,
.
numebr on his list? Write its last two digits as your answer.
grid of square tiles. Robert chooses four of these squares uniformly at random. The probability that the centers of these four squares form a square themselves is 
where 
are coprime naturals. Write the largest 
digit odd factor of 
.
.
which touches one face of the cube and passes through all 
for all x, y 
0, where 
is the prime number.