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.
such that 
are four given points on the plane, so that no three of them are collinear. Let 
be the set of conics passing through . Consider a fixed point 
, for each 
, suppose there are distinct points 
, so that 
. Let 
be the line joining the intersection of 
and 
with the intersection of 
and 
. does not depend on the choice of . passes through a fixed point when 
varies."
such that:![\[f(1+xf(y))=yf(x+y)\]](http://latex.artofproblemsolving.com/f/2/a/f2a9e8eb184c921d5f1fa59137d99e38b29c6962.png)
be an acute scalene triangle , 
be the orthogonal projection of 
on 
, 
and 
are the midpoints of 
and 
respectively. Let and 
are points on the minor arcs 
and 
of the circumcircle of triangle respectively such that 
. Show that the circumcircles of triangles 
and 
are tangent if and only if lies on 
.
, where 
, and 
are positive integers that satisfy
and 
.
be real positive numbers such that: 
. Prove that: 
: 
such that:
for all x,y E 
, 
, 
, 
be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, 
. Prove that 
for all 
.
with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points 
with integer coordinates such that 
for every pair 
be a trapezoid inscribed in circle 
, 
. Let is the symmetric point of across , 
intersects at 
. Let is midpoint of and 
is orthocenter of triangle . On 
take a point 
so that 
. Prove that: 
coincide
be integer numbers 
satisfy that:
. Prove that: are even
is a perfect square number
can’t be any power 
of a positive integer number
and of an equilateral triangle , points and are chosen such that 
. Find 
, if is the intersection point of the segments 
and 
.
and of an equilateral triangle , points and are chosen such that . Find , if is the intersection point of the segments and .
and 
Find the minimum 
Prove that![$$\frac{a}{b}+ \frac{kb^2}{c^2} + \frac{c}{a}\geq 5\sqrt[5]{\frac{k}{16}}$$](http://latex.artofproblemsolving.com/8/b/e/8bed6605f3988f59b8395b091b92a8b9e8b40419.png)
Where 
