[/quote]
,
,
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be the roots of 
with 
and 
.![$[|3 \alpha |] + [|6 \beta|] +
[|9 \gamma |]$](http://latex.artofproblemsolving.com/9/f/f/9ffb5118d94f0e2fb84a411b47ec91c122985673.png)
is equal to
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 
and 
are the two intersections of the perpendicular from 
with respect to side 
with the circle of diamter 
and 
lies on 
.
, but no more than 
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 
.
,
rows in which the majority of cells are blue, and exactly 
such that![\[
2^a \cdot 3^b + 1 = 5^c.
\]](http://latex.artofproblemsolving.com/0/a/c/0ac1f22162eacf698847bceb33f9a337bd53c44b.png)
be an equilateral triangle with circumcenter 
.
and 
be two points on segments 
and 
,
.
is the reflection of 
,
and 
are parallel.
from the set of positive integers to the set of positive integers such that, for all positive integers 
and 
,![\[ a, f(b) \text{ and } f(b + f(a) - 1).\]](http://latex.artofproblemsolving.com/4/a/b/4ab97da27e05d0f9e7762f74d0d0bd2dc8b9842f.png)
be a cyclic pentagon inscribed in a circle of centre 
which has angles 
.
and 
meet at a point belonging to the diameter 
.
its circumcircle and 
intersects 
and the segment 
in a point 
.
be the midpoint of 
.
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)
.
Y by
Hey AoPS!
I came across a problem recently and it goes like this:
Which is greater:
or 
Soo... can u guys help? thx!
and may the force be with u!
Notes: I don't exactly know where this problem came from, but if u find that u are the orig maker of this problem feel free to drop me a PM and ill add u to my post
I came across a problem recently and it goes like this:
Which is greater:
or 
Soo... can u guys help? thx! and may the force be with u!Notes: I don't exactly know where this problem came from, but if u find that u are the orig maker of this problem feel free to drop me a PM and ill add u to my post
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