[/quote]
,
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 an acute-angled triangle with 
.
be the circumcircle of 
be the midpoint of the arc 
of 
.
meets 
at 
and meets 
.
at 
.
by 
.
.
meets line 
.
be a cyclic pentagon inscribed in a circle of centre 
which has angles 
.
and 
meet at a point belonging to the diameter 
.
and the segment 
intersects the circumcircle of 
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)
.
from the nonnegative integers to the integers satisfying 
and ![\[ f(mn) = f(m) + f(n) + f(m)f(n) \]](http://latex.artofproblemsolving.com/e/f/f/eff10a32cffaec742403ebfc46ff09bc8ef7b7c5.png)
for all nonnegative integers 
and 
.
,
,
![\[ \frac {f(x) + f(y)}{2} \ge f\left(\frac {x + y}{2}\right) + |x - y|
\]](http://latex.artofproblemsolving.com/3/b/a/3ba7733caf51cc70a090920ca21f8b47aad9b1ec.png)
and 
.
satisfying
for all integers 
.
,
,
means 
means 
applied 
Y by
In the accompanying figure, semicircles with centers
and 
have radii 
and 
, respectively. Furthermore, they are internally tangent to the circle of diameter 
. Also the semicircles with centers and 
are externally tangent to each other. The circle with center 
is internally tangent to the semicircle with diameter and externally tangent to the others two semicircles. Determine the value of the radius of the circle with center .
and 
have radii 
and 
, respectively. Furthermore, they are internally tangent to the circle of diameter 
. Also the semicircles with centers and 
are externally tangent to each other. The circle with center 
is internally tangent to the semicircle with diameter and externally tangent to the others two semicircles. Determine the value of the radius of the circle with center .This post has been edited 1 time. Last edited by parmenides51, Oct 2, 2021, 1:41 PM
Therefore 
.
