AoPS Academy
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, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
, and 
are all polynomials such that ![\[ P(x^5)+xQ(x^5)+x^2R(x^5)=(x^4+x^3+x^2+x+1)S(x),\]](http://latex.artofproblemsolving.com/1/6/a/16a71abc110ece558d427a07d7be2d1f72148024.png)
prove that 
is a factor of 
.
be a point in the plane, and 
a line not passing through . Evan does not have a straightedge, but instead has a special compass which has the ability to draw a circle through three distinct noncollinear points. (The center of the circle is not marked in this process.) Additionally, Evan can mark the intersections between two objects drawn, and can mark an arbitrary point on a given object or on the plane. over ? to ?
is a cyclic quadrilateral with 
. Let 
and 
be the feet of perpendiculars from and 
to 
respectively. Prove that 
.
of positive integers is called central if for every positive integer 
, the arithmetic mean of the first 
terms of the sequence is equal to .
of positive integers such that for every central sequence 
there are infinitely many positive integers with 
.
. Prove that :
be positive integers. If for each positive integer we have that 
Prove that 
.
cats consisting of five different breeds. He records the number of cats of each breed and after multiplying these five numbers he obtains the number 
. How many cats of each breed does he have?
satisfying 
Prove that 
, if any element of the set 
is fantabulous, then all the elements are fantabulous. Does it follow that the number 
is fantabulous?
such that:
and 
: Prove that :
and 
Prove that
is 
,
with the centroids of the opposite faces are equal to 
and 
,
