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.
and 
intersect at points 
and 
. A line through intersects and at points 
and 
, respectively. Line 
intersects at point 
, and line 
intersects at point 
. If 
is the circumcenter of 
, prove that 
.
be a positive integer. Initially the sequence 
( times) is written on the board. In each round, Ananya choses an integer 
and a subset of the numbers written on the board and adds to all of them. What is the minimum number of rounds in which Ananya can make the sequence on the board strictly increasing?
be real numbers . Prove that : There exist positive integer 
such that 
Where 
, 
, 
are pairwise distinct real numbers satisfying 
Prove that
Let , , are pairwise distinct real numbers satisfying 
Prove that
be a triangle with incenter 
, and let be a point on line 
satisfying 
. Let the excircle of triangle opposite the vertex be tangent to 
at 
. Define points 
on 
and 
on 
analogously, using the excircles opposite and , respectively.
is cyclic, then 
is tangent to the circumcircle of 
.
, consider the following parabola on the coordinate plane.
ranges over all positive real numbers, draw the domain of the set swept out by .
be a triangle with circumcircle 
and incenter and let 
be the midpoint of . The points , 
, 
are selected on sides , , such that 
, 
, and 
. Suppose that the circumcircle of intersects at a point 
other than . Prove that lines 
and 
meet on .
and 
touches each other externally at a point 
, quadrilateral 
is inscribed in , and the lines 
, 
are tangent to at points and respectively. Line 
bisects 
and meets segment 
at 
. Line 
meets the arc 
(not passing through the point ) at another point different from . Prove that is the circumcenter of 
.
be an acute triangle with circumcenter and orthocenter 
, satisfying 
. The tangent line at to the circumcicle of intersects in . Let be the midpoint of 
. Prove that 
.
be the incentre of acute-angled triangle . Let the incircle meet 
, and 
at 
, and 
respectively. Let line intersect the circumcircle of the triangle at 
and 
, such that lies between and . Prove that 
. such that 
. The lines and intersect at a point and the line passing through which is parallel to , intersects 
at . Prove that
be the circumcircle of acute triangle . Two tangents of from and intersect at , 
and intersect at . Point , are on and such that 
and 
.
are concyclic.
the centre of the circle passing through . 
, 
are difined similarly. Prove that 
, 
, 
are concurrent.
be a fixed positive integer. Given a set 
of 
points in the plane such that no three are collinear and no four concyclic, let 
be the number of circles 
that contain 
in their interior, and let ![\[m(S)=a_1+a_2+\cdots + a_n.\]](http://latex.artofproblemsolving.com/4/9/b/49b928b1f20e2d1799a1234c7f25a2bc3d62d0dd.png)
Prove that there exists a positive integer 
depending only on 
such that the points of 
are the vertices of a convex polygon if and only if 
and the altitude by 
We are asked to show that 
However, 
where 
Also, we know that 
Therefore, 
is equivalent to
by the statement of the problem we have 
and 
Thus, 
is equivalent to 
which can be rewritten as (1-x)(1-y)+(1-x)(1-y)(x^3+y^3+5xy-5x-5y+4)\ge 0.\ (3)$](http://latex.artofproblemsolving.com/c/9/2/c925228ce8276a896698f351889bd6457b4052d2.png)
because all other terms in 
are 
But for 
this is obvious because ![$f(x,y)=(x+y-1)^3+(1-x)(1-y)\left[5-3(x+y)\right]\ge 0,$](http://latex.artofproblemsolving.com/f/0/1/f0150f9a8e2d45f6f70752db8f625b66ff384b78.png)
while for 
we have 
holds under the conditions of the problem. Equality occurs when the triangle is an equilateral triangle.
we have 
and 
Prove that the ratio of circumcircle radius to incircle diameter is at least as big as the length of the centroidal axis 
to the altitude 
When do we have equality?