AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be a acute triangle with altitudes 
. 
is the orthocenter of . The angle bisectors of 
intersect 
at 
, respectively. The circle 
with diameter intersects segment 
at 
. Show that is tangent to 
.
with real coefficents, that satisfies 
for all real numbers 
is inscribed in circle 
. The interior angle bisector of angle 
intersects side and at 
and 
(other than ), respectively. Let 
be the midpoint of side . The circumcircle of triangle 
intersects sides 
and 
again at 
and 
(other than ), respectively. Let 
be the midpoint of segment 
, and let be the foot of the perpendicular from to line 
. Prove that line 
is tangent to the circumcircle of triangle 
.
be a scalene triangle, and let 
be its incenter. A point is chosen on line , such that the circumcircle of triangle 
intersects at 
, and the circumcircle of triangle 
intersects at 
. Circumcircle of triangle 
intersects and at and , respectively. Lines 
and 
intersect at , and lines 
and 
intersect at . Prove that 
.
. Solve the equation: 
positive integers are written on a board. It is known that, after you erase a number from the board, there exists a number that is equal to the arithmetic mean of the rest of the numbers left on the board. Find all the numbers that could've been erased.
is a fifth degree polynomial. 
, 
, 
, 
. 
and a point 
outside the circle is given. Let a tangent from to touch at , and take points 
lying on such that 
are colinear. The bisector of 
intersects and at and ,respectively. Prove that 
. If 
, then prove the following: 
. Prove the following inequality: ![$$\sqrt[3]{\frac{a}{b}}+\sqrt[3]{\frac{b}{a}}\leq\sqrt[3]{2(a+b)(\frac1{a}+\frac1{b})}$$](http://latex.artofproblemsolving.com/7/0/7/70776d13de2e13b8075bd1bd6883a8968426645a.png)
, which lie on a circle, are marked by distinct reals 
. Let 
be 
good for a on the circle different than , if and only if 
is the greatest number on at least one of the two arcs 
. Let the score of be the number of good points on the circle. Determine the greatest 
such that regardless of the values of , there exists a point with score at least .
are written on a board. and 
plays the following game: They take turns choosing a number from the board and deleting them. starts first. They sum all the deleted numbers. If after a player's turn (after he deletes a number on the board) the sum of the deleted numbers can't be expressed as difference of two perfect squares,then he loses, if not, then the game continues as usual. Which player got a winning strategy?
has a right angle at 
. The internal bisectors of angles 
and meet and 
at and respectively. The points and are the feet of the perpendiculars from and to . Find angle 
.
has a right angle at . The internal bisectors of angles and meet and at and respectively. The points and are the feet of the perpendiculars from and to . Find angle .
concyclic , 
concyclic
.
is a cyclic quadrilateral and quadrilateral 
is a cyclic quadrilateral
, we get 