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with real coefficients of degree greater than 
is given. Prove that there are infinitely many positive integers which cannot be represented in the form ![\[f(n+1)+f(n+2)+\cdots+f(n+k)\]](http://latex.artofproblemsolving.com/a/6/4/a648623198555fa749157c99a83eec8e74c98961.png)
where 
and 
are positive integers.
+1 w
satisfies 
. Find the value of 
.
be a scalene triangle with incenter 
and incircle 
. Let the tangency points of to 
be 
respectively. Let the line 
intersect the circumcircle of at the points 
. Assume that 
lies between the points 
and 
. Let 
be a circle that passes through and 
and that is tangent to at the point 
which lies on different semi-planes with 
with respect to the line . Let intersect 
at points 
and 
and let the second intersection point of the circumcircle of and the circumcircle of 
be 
. Prove that the intersection point of 
and 
lies on the circumcircle of if and only if the intersection point of 
and 
lies on .
a tangencial quadrilateral that is not a rhombus, let 
be lengths of tangents from 
to the incircle of the quadrilateral which center is . Let 
be the midpoints of 
resp. Prove that![\[ \frac{MI}{IN}=\frac{a+c}{b+d} \]](http://latex.artofproblemsolving.com/e/7/0/e709acb8b17c0466a385ba8e7a89bb1bf16c72af.png)
