AoPS Academy
with 
odd, which satisfy the equation:
and 
are two sequences between 
and 
and they satisfy 
, prove that 
, and determine when does equality hold
have the property that there exists an infinite sequence 
of nonzero integers such that the equality ![\[a_k+2a_{2k}+\ldots+na_{nk}=0\]](http://latex.artofproblemsolving.com/5/b/a/5ba1051daa520af4e7a53ece725c400b229075cf.png)
holds for every positive integer 
.
be a given point inside quadrilateral 
. Points 
and 
are located within such that![\[\angle Q_1BC=\angle ABP,\quad\angle Q_1CB=\angle DCP,\quad\angle Q_2AD=\angle BAP,\quad\angle Q_2DA=\angle CDP.\]](http://latex.artofproblemsolving.com/d/b/3/db31d11ec24322724c0e0f1485aadf8864eabf7a.png)
Prove that 
if and only if 
.
from each of the integers at two neighboring vertices and adding 
to the opposite vertex, which is not adjacent to either of the first two vertices. (The amount and the vertices chosen can vary from turn to turn.) The game is won at a certain vertex if, after some number of turns, that vertex has the number 2011 and the other four vertices have the number 0. Prove that for any choice of the initial integers, there is exactly one vertex at which the game can be won.
be the largest real solution to the equation ![\[\frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}= x^2-11x-4.\]](http://latex.artofproblemsolving.com/e/8/7/e87fdc9fe63d29a46e57dfb92c2d0448441b4b55.png)
There are positive integers 
such that 
. Find 
.
is harmonic where 
is the intersection of 
with 
where R is the intersection of the angle bisector of angle C with 
and 
is the intersection of 
with 
Since 
is 
degrees and 
bisects 
we are done.
are parallel, then the midpoints of and 
are collinear (which follows from similar triangles).
to . Let be the 
-intouch point, 
the antipode on the incircle, 
the -excircle touch point, the -excircle antipode, and 
the reflection of 
over 
. By homotheties at the incircle and excircle, 
are collinear. Then this implies that 
are collinear since 
is collinear. Finally, since 
are collinear, it follows that since 
that 
are collinear as desired.
the antipodal of on 
then on 
invert we have 
and 
and combining I-E Lemma with the fact that the dimater makes a 90 degree we have 
which means that 
so as 
we are done.
and 
Prove that 
with Incenter 
and A-Excenter 
,
