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 convex quadrilateral such that diagonals 
and 
intersect at right angles, and let 
be their intersection. Prove that the reflections of across 
are concyclic.
be a triangle with circumcircle 
and incentre 
. Let the line passing through and perpendicular to 
intersect the segment 
and the arc (not containing 
) of at points 
and 
, respectively. Let the line passing through and parallel to 
intersect 
at 
, and let the line passing through and parallel to intersect 
at 
. Let 
and 
be the midpoints of 
and , respectively. Prove that if the points 
and are collinear, then the points 
and are also collinear.
, prove
In think this is interesting so I pose it here.
be a triangle with circumcenter 
,orthocenter 
is the radius of 
. lie on the plane such that 
and 
are not in the same half plane of .
lie on such that 
and farthest to .
intersects the perpendicular bisector of 
at 
. 
intersects again at 
intersects at 
lie on 
such that 
. 
is perpendicular to 
at 
. Prove that: 
is the midpoint of 
.
be the set of all non-decreasing functions 
satisfying 
for all positive integers 
. Find the maximum value of
over all 
. with (I) is incircle and 
is the Gergonne point. 
and 
is the midpoint of 
The perpendicular line from to meet 
at 
Draw diameter 
of 
Prove that is a point on the polar of through 
be a Cassinian oval with center and foci 
, 
whose Cartesian equation is expressed as follows:
Let 
be a point on and 
the reflection in of .
be a circle passing through and , and tangent to at a point 
different from , ., 
be the second intersection points of 
, 
with , respectively. be the orthocenter of 
, and 
the orthogonal projection of onto 
., , , , , are conconic.
be an acute-angled triangle with 
and let be the centre of its circumcircle 
. Let 
be a point on the line segment such that 
. Let 
be the second point of intersection of and the line 
. If , and are the midpoints of the line segments 
, 
and 
, respectively, show that the points , and are collinear.
be a nonempty subset of the points in the Cartesian plane such that for each 
exactly one of 
or 
also belongs to . Prove that for each positive integer 
there is a line in the plane (possibly different lines for different ) which contains at least points of .
be a triangle. The incircle of touches the sides and at the points and , respectively. Let be the point where the lines 
and 
meet, and let and be points such that the two quadrilaterals 
and 
are parallelogram.
.
. Find all functions 
, such that for any positive reals 
, 
,![\[f\left(f(x)^2+\frac 13yf(y)\right)=x^2-f^{(m)}(x)+3x+(m-n+2)+f^{(n)}\left(\frac 13(y^2+y)\right).\]](http://latex.artofproblemsolving.com/a/8/0/a80871386dbd14de02a18b521c702564b95cffe3.png)
Created by Fanyu Meng
, 
is the circumcircle and 
is the nine-point circle. The tangents to at 
and 
intersect at 
. Let 
be a point on such that 
and does not lie on 
. Construct a circle 
symmetric to with respect to , and let intersect at points 
and 
. Let 
be the isogonal conjugate of with respect to . Prove that 
are concyclic.
, and an interval glued with one of its endpoints to the circle.
and a positive integer 
is given.
and for each 
![\[f_n(k)=\frac{a_n+\dots+a_{k+1}}{n-k}-\frac{a_k+\dots+a_1}{k}\]](http://latex.artofproblemsolving.com/0/4/1/041f80cf63f17c0f62c166021af474a5d305eb87.png)
Then for every integer 
the condition![\[a_{n+1}=\max_{0\leq k<n}(f_n(k))\]](http://latex.artofproblemsolving.com/b/1/f/b1f5aff787de8fc9c113cf8441a6b2900a19f24f.png)
is satisfied. Prove that this sequence must be eventually constant.
and a positive integer is given. and for each Then for every integer the conditionis satisfied. Prove that this sequence must be eventually constant.
, prove 
.
. Also 
. We have 
. We will proceed by using induction. Suppose that 
holds, we will prove 
. On the other hand, for all 
Thus, we obtain the following
implying that the sequence 
must be eventually constant as desired.
![\[a_{n+1}=\max_{0\leq k<n}(f_n(k))=f_n(g(n))\]](http://latex.artofproblemsolving.com/8/b/1/8b11c506616b015473f027f4874c01d88634da4e.png)
Step 
:
,
,
will hold.
: