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.
points and some chords between them is given.
In every step we can take two chords 
with a common point other than 
and erase exactly one of and draw 
let 
be the minimum of chords after some steps. Find the maximum of over all initial positions.
In every step we can take two chords with a common point other than and erase both of and draw let be the minimum of chords after some steps. Find the maximum of over all initial positions.
be a positive integer and let 
and 
be positive integers congruent to 1 modulo 4. Prove that there exists a positive integer 
such that at least one of the numbers 
and 
is divisible by 
be the lengths of the sides of a triangle. Prove that![\[\frac{1}{(a-4b)^2}+\frac{1}{(b-4c)^2}+\frac{1}{(c-4a)^2} \geqslant \frac{1}{ab+bc+ca}.\]](http://latex.artofproblemsolving.com/7/4/b/74b7aa5621bc52977d9acd448d74293dc27a8e00.png)
and 
be arbitrary points on the sides 
and 
of triangle 
, respectively. The circumcircle of 
meets for the second time the circumcircle of 
at point 
. Line 
meets line 
at point 
, while line 
meets line 
at point 
. Prove that lines 
and 
pass through the same point.
with circumcenter 
. 
and 
intersect the altitude from 
to at points 
and 
respectively. 
is the circumcenter of triangle 
and 
is the reflection of over . 
is the second intersection of circumcircles of triangles 
and 
. Show that 
are collinear.
and 
Find the value of 
and 
such that 
and 
is a prime number
such that 
for all 
be an even integer and let 
be real numbers satisfying 
.
be a positive integer. Ana and Banana play a game. Banana thinks of a function 
and a prime number 
. He tells Ana that 
is nonconstant, 
, and 
for all integers 
. Ana's goal is to determine the value of . She writes down integers 
. After seeing this list, Banana writes down 
in order. Ana wins if she can determine the value of from this information. Find the smallest value of for which Ana has a winning strategy.
Let be positive reals such that 
.Prove the inequality 
be a quadrilateral. The incentres of the triangles 
form a rectangle. Proove, that is cyclic.
,
,
are the vertices of a rectangle.
such that 
,![\[I_{ABC}=-bc-ca-ab,I_{BCD}=-cd-db-bc,I_{CDA}=da+ac-cd,I_{DAB}=-ab+bd+da.\]](http://latex.artofproblemsolving.com/e/c/4/ec411eef7e840f61ad197bb0c5a5d4f3acac8073.png)
Then we can compute ![\[I_{ABC}+I_{CDA}=-ab-bc-cd+da=I_{BCD}+I_{DAB}\]](http://latex.artofproblemsolving.com/e/d/2/ed26d66b70826c1d667729141f08cbae4dee880b.png)
and
so the incenters form a rectangle.
is cyclic. And so is 
.
.