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with diameter 
is given. The point 
lies in the interior of the circle, but not on . The line 
intersects in 
and 
. The tangent to at intersects the line through perpendicular to , at 
. The point 
lies on , and is such that 
is tangent to and 
.
, and are collinear.
has centre 
, and 
is a diameter of . Let be a point of such that 
. Let 
be the midpoint of the arc 
which does not contain 
. The line through parallel to 
meets the line 
at 
. The perpendicular bisector of 
meets at 
and at 
. Prove that is the incentre of the triangle 
lies inside triangle 
such that 
and 
. Point is the midpoint of segment . Point lies on segment with 
. Prove that 
.
be a function such that 
for all pairs 
of positive integers. Prove that there exists a positive integer 
which divides all values of 
.
, points and 
are taken on the legs and , respectively, so that 
and 
. Find the acute angle between line segments 
and 
.
. Find the max 
that 
for all 
.
be real numbers such that ![\[ \frac{1}{x^2-1} + \frac{1}{y^2-1} + \frac{1}{z^2-1} = 1. \]](http://latex.artofproblemsolving.com/8/f/5/8f5d49c9b44e3db3cd85fea7b6521e1dc02d4269.png)
Prove that ![\[ \frac{1}{x+1} + \frac{1}{y+1} + \frac{1}{z+1} \le 1. \]](http://latex.artofproblemsolving.com/c/0/4/c0438d6eba686d0a11fd4decb4afa9a14aa50bac.png)
be real numbers. Prove that
where the indices are taken modulo 8, i.e., 
, 
, and 
. In which cases does equality hold?
be a circumscribed quadrilateral with incircle 
and no two opposite angles equal. Let be an arbitrary point on the diagonal 
, which is inside . The segments 
and 
intersect at 
and . The tangents to at and intersect at . Prove that lies on the line .
be an acute triangle with 
, midpoint of side , altitude 
(
), and orthocenter 
. A circle passes through points 
and 
, is tangent to line 
, and intersects the circumcircle of triangle at a second point 
. The circumcircle of triangle 
intersects line 
at a second point 
. Prove that the lines 
and 
are perpendicular.
be a convex quadrilateral such that 
. Suppose that a point on the side 
satisfies the equality![\[AE\cdot ED + BE^2=CD\cdot AE.\]](http://latex.artofproblemsolving.com/d/7/5/d75cc3a449a56b43eac574cc44e9932a2e63b33b.png)
.
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 .
and 
.Find max 
.
for all 
be real numbers. Prove that
be two lines .
s.t. 
and the newton line of 
is parallel to 