AoPS Academy
be a triangle with points 
and 
lying on rays 
and 
, respectively, such that 
. On the line 
, let points 
and 
be chosen so that 
and 
. Prove that the circumcenter of triangle is equidistant from the midpoints of segments 
and 
., for which 
, the radius of the inscribed circle is 
, the midpoint of its arc 
(which does not contain 
) is 
. A point 
is placed on the extension of altitude 
such that 
is between 
and and 
. Prove that 
.
, 
, 
, 
, 
in some order.
of positive integers such that ![\[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\]](http://latex.artofproblemsolving.com/e/c/7/ec778edf97100f3b19350d8a817317ebf1f391bd.png)
+1 w
is said to be good if 
is not the square of an integer. Determine all integers 
with the following property: can be represented, in infinitely many ways, as a sum of three distinct good integers whose product is the square of an odd integer.
of positive integers such that 
and
holds for every positive integer 
.
, your right-neighbor’s is 
, and the next person over is 
, then for every trio in the circle they see
and polynomial ![$P(x) \in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/4/5/3/453a624c3b002c1b0e78e0023b24dd22ddd03557.png)
with nonnegative coefficients such that![\[f(x + P(x)f(y)) = (y + 1)f(x)\]](http://latex.artofproblemsolving.com/0/b/4/0b49f2379b5fae0489a5a9ab9a890de5c3b679dd.png)
for all 
.
the orthocenter of . Construct line 
through perpendicular to 
. Construct the line through perpendicular to 
and let it intersect at 
. Let the line though perpendicular to 
meet 
respectively at 
. These are the required points. the orthocenter of 
and using that we finally constructed 
,the miquel point of 
is the infinity point of the newton line so we can construct 
then we draw a line parallel to through 
; it hits 
at another point say 
let the line 
cuts 
at point that is exactly . Finally we draw a parallel to to construct .
and two vectors 
, construct a line , with direction 
, such that Newton line of the quadrilateral formed by the four lines 
has direction 
. form a triangle . Let 
be the parabola inscribed in its medial triangle and whose axis has the direction of the vector 
. Let be the conjugate diameter of the direction of the vector . and intersects at a point 
. The tangent at to is Newton's line of the quadrilateral 
, which has the direction of the vector .
be two lines .
s.t. 
and the newton line of 
is parallel to 