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 circle, and 
are two fixed points on . Given a third point 
on , let 
and 
denote the feet of the altitudes from 
and 
, respectively, in the triangle 
. Prove that there exists a fixed circle 
such that 
is tangent to regardless of the choice of the point .
Choose a natural number 
such that 
. If both numbers and 
are on the board, erase both.
Choose a natural number such that 
. If both numbers and 
are on the board, erase both.
be real numbers satisfying 
Prove that 
and determine all cases of equality.
be a scalene acute triangle with incentre 
and circumcentre 
. Let 
cross 
at 
. On circle , let and be the mid-arc points of and 
, respectively. Let 
cross 
at 
and let 
cross 
at 
. Prove that the lines 
and 
are concurrent on the external bisector of 
.
+1 w
be a function from the set of integers to the set of positive integers. Suppose that, for any two integers 
and 
, the difference 
is divisible by 
. Prove that, for all integers and with 
, the number 
is divisible by 
.
, let 
and 
be a pair of isogonal conjugate points. The line 
intersects 
at 
, and the line 
intersects at 
. Let the circumcircle of 
and the circumcircle of 
intersect again at 
(other than ). Prove that the line 
bisects 
.
board black or white so that the cells of each color form a polygon. Then Bob dissects the board into dominoes (rectangles consisting of two cells). Alice wants to maximize the number of two-colored dominoes, and Bob wishes to minimize it. What maximal number of two-colored dominoes can be guaranteed by Alice regardless of Bob’s moves? (Recall that the boundary of a polygon is a closed broken line without self-intersections.). If![\[|a+b+c|=|ab+bc+ca|=|abc|=1,\]](http://latex.artofproblemsolving.com/b/6/7/b6762ca030e77f37d385db6ebf0de03cff3799f5.png)
show that 
.
, 
, 
of an acute-angled triangle intersect at point 
. The bisectors of angles and of triangle 
meet the segments 
and 
at points and respectively. Denote the value of the angle 
by 
. Define 
and 
similarly. Find the sum 
. with incircle 
tangent to 
at points 
, respectively. Let 
be a point such that its isogonal conjugate lies on the line 
(where 
is the circumcenter and 
the incenter of ). The line 
intersects segments 
and 
at points 
and 
, respectively, such that the circle with diameter 
meets at points 
and 
.
is tangent to the incircle at some point .
corresponding to vertices . Prove that the lines 
are concurrent.
such that : 
let 
be the circumcircle radius of the triangle 
If 
is a triangle with incircle centre 
then we have:![\[ \frac{1}{R_{ABI}} + \frac{1}{R_{BCI}} + \frac{1}{R_{CAI}} \leq \frac{1}{\bar{AI}} + \frac{1}{\bar{BI}} + \frac{1}{\bar{CI}}.\]](http://latex.artofproblemsolving.com/0/8/4/0841e84d09160d41dd2cb232ad23d88747428d48.png)
and 
,
.
and 
,
)