[/quote]
,
and 
.
be an acute triangle with orthocenter 
and circumcenter 
.
and 
be points on the line segments 
and 
respectively such that 
is a parallelogram. Prove that 
.
be the midpoint of side 
.
intersect 
at 
.
is tangent to 
,
.
,
be the orthocenter of it and 
be any point on the side 
.
are on the segments 
,
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
.
and 
and 
are positive integers, we say that 
and 
.
such that 
,
and 
,![$\sqrt{ad^3 g^4} +\sqrt[3]{be^3 h^4} + \sqrt[8]{cf^3 k^4}$](http://latex.artofproblemsolving.com/b/f/a/bfa17e8bb4cc05a84ed6a757f506d6d0ed2b0081.png)
is an odd number, let 
be a positive integer. If 
then 
.
points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least 
scalene triangles with vertices of that color.
and 
.
?
be positive integers satisfying ![\[\frac{ab}{a+b}+\frac{cd}{c+d}=\frac{(a+b)(c+d)}{a+b+c+d}.\]](http://latex.artofproblemsolving.com/b/8/c/b8c3fe14640a2dcb30b755b2347721da5d080a78.png)
Determine all possible values of 
.
users, some pairs of whom are friends. Whenever user 
is friends with user 
users have 
friends each, and 
and natural 
Y by
In a school there are 
classes and 
student. We know that in this school every two students have attended exactly in one common class. Also due to smallness of school each class has less than students. If 
is not a perfect square, prove that there exist a student that has attended in at least 
classes.
Proposed by Mohammad Moshtaghi Far, Kian Shamsaie Rated 4
classes and 
student. We know that in this school every two students have attended exactly in one common class. Also due to smallness of school each class has less than students. If 
is not a perfect square, prove that there exist a student that has attended in at least 
classes.Proposed by Mohammad Moshtaghi Far, Kian Shamsaie Rated 4
This post has been edited 3 times. Last edited by matinyousefi, Mar 12, 2020, 6:10 PM
classes, say they are 
for some student 
.
be the set of students 
among the 
share 
.
,
students. Now consider some student 
outside of this class. This 
classes, contradiction. Therefore some student attends 
classes. Since 
classes.
