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continuous function with 
for all real 
.
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 
.
, consider the generating function 
. Clearly, this yields 
. Therefore, we obtain
, we have
, it is straightforward to observe that the coefficient of 
in 
is 
. This leads to the conclusion.
, consider the generating function . Clearly, this yields . Therefore, we obtain, we have, it is straightforward to observe that the coefficient of in is . This leads to the conclusion.
and 
Prove that 
satisfying:
and 
where each summation runs over 
variables 
Then find the value of 
in 
is 
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