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.
to be an acute triangle. Also, let 
and 
to be the two intersections of the perpendicular from 
with respect to side 
with the circle of diameter , with closer to than . Analogously, 
and 
are the two intersections of the perpendicular from 
with respect to side 
with the circle of diamter , with closer to than . Prove that the intersection of 
and 
lies on 
.
satisfying![\[f(x + yf(x)) + f(xy) = f(x) + f(2019y),\]](http://latex.artofproblemsolving.com/8/0/f/80fee0440e9fc4031fa4dcff96b9605011ef0e68.png)
for all real numbers 
and 
.
, in which each unit cell is colored blue or red. There are exactly 
rows in which the majority of cells are blue, and exactly columns in which the majority of cells are red.
be an equilateral triangle with circumcenter 
. Let 
and 
be two points on segments 
and 
, respectively, such that 
. If 
is the reflection of with respect to line 
, prove that lines 
and 
are parallel.
from the set of positive integers to the set of positive integers such that, for all positive integers 
and 
, there exists a non-degenerate triangle with sides of lengths![\[ a, f(b) \text{ and } f(b + f(a) - 1).\]](http://latex.artofproblemsolving.com/4/a/b/4ab97da27e05d0f9e7762f74d0d0bd2dc8b9842f.png)
be a cyclic pentagon inscribed in a circle of centre 
which has angles 
. Show that the diagonals 
and 
meet at a point belonging to the diameter 
. be an acute triangle, 
its circumcircle and its circumcenter. The altitude from 
intersects in a point 
and the segment intersects the circumcircle of 
in a point 
. Finally, let 
be the midpoint of 
. Show that 
is parallel to 
.
be the set of all nonnegative integers. Find all the functions 
satisfying the relation![\[ f(f(f(n))) = f(n+1 ) +1 \]](http://latex.artofproblemsolving.com/8/b/9/8b953428105df2c2344963cbaf3eb81d8201e0df.png)
.
such that![\[
2^a \cdot 3^b + 1 = 5^c.
\]](http://latex.artofproblemsolving.com/0/a/c/0ac1f22162eacf698847bceb33f9a337bd53c44b.png)
