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be an acute triangle and 
its circumcircle. Let 
be the line tangent to at 
. Let 
and 
be the intersections of the circumference with center 
and radius 
with lines and 
, respectively. Prove the orthocenter of lies on line 
.
be a cyclic quadrilateral an let 
be a point on the side 
The diagonals meets the segments 
at 
The line through parallel to 
mmets the extension of the side 
beyond at 
The line through 
parallel to 
meets the extension of the side beyond at 
Prove that the circumcircles of the triangles 
and 
are tangent .
be an integer greater than 
. For a positive integer 
, let 
. Suppose that there exists a 
-element set 
such that is an -element subset of 
; shares at most one common element; is contained in exactly two elements of . in terms of .
be an integer and 
be the focus of the parabola 
. A regular polygon 
has the center in and none of its vertices lie on 
. 
intersect the parabola at 
.![\[ FB_1 + FB_2 + \ldots + FB_n > np . \]](http://latex.artofproblemsolving.com/5/a/8/5a8a7817db9c16e18c90209bd9b0a1df228cbb20.png)
be an integer and be the focus of the parabola . A regular polygon has the center in and none of its vertices lie on . intersect the parabola at .
.![\[ B_i: (\frac{p}{2}+FB_i\cdot{cos(\phi+(i-1)\frac{2\pi}{n})}, FB_i\cdot{sin(\phi+(i-1)\frac{2\pi}{n})}) \]](http://latex.artofproblemsolving.com/8/2/3/823566e0964dab20d226868c5f78203a0752c194.png)
belongs to the parabola it yields that:![\[ FB_i=\frac{p}{1-cos(\phi+\frac{2\pi}{n}(i-1))} \]](http://latex.artofproblemsolving.com/5/c/d/5cd7608dbabfbacd11781d4764c819596e70e52f.png)
![\[ \sum_{i=1}^{n}{\frac{1}{FB_i}}=\frac{n}{p} \]](http://latex.artofproblemsolving.com/b/3/5/b350867dad54d201bb7e24f8757ad517c414c9ca.png)