[/quote]
,
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 
.
.
be the circumcircle of 
be the midpoint of the arc 
of 
.
at 
and meets 
.
.
by 
.
.
.
,
,
very convex if![\[ \frac {f(x) + f(y)}{2} \ge f\left(\frac {x + y}{2}\right) + |x - y|
\]](http://latex.artofproblemsolving.com/3/b/a/3ba7733caf51cc70a090920ca21f8b47aad9b1ec.png)
and 
.
such that![\[
2^a \cdot 3^b + 1 = 5^c.
\]](http://latex.artofproblemsolving.com/0/a/c/0ac1f22162eacf698847bceb33f9a337bd53c44b.png)
satisfying
for all integers 
.
,
,
means 
means 
applied 
satisfying 
.
such that 
(the number of positive divisors of 
be integers and 
be a prime for which![\[ x^2-3xy+p^2y^2=12p \]](http://latex.artofproblemsolving.com/2/4/c/24cb51eb55144a9a1e0e489d115993496a052e79.png)
.
satisfying 
for all positive integers 
be an isosceles trapezoid with 
.
be the midpoint of 
.
and 
,
is tangent to 
Y by
Let 
be an acute triangle. Let us denote the foot of the altitudes from the vertices 
and 
to the opposite sides by 
and 
respectively, and the intersection point of the altitudes of the triangle 
by 
Let 
be the intersection of the line 
and the segment 
A straight line passing through and perpendicular to 
intersects 
at 
Let 
be the intersection of the segment 
with the perpendicular drawn from 
Prove that is the midpoint of segment 
be an acute triangle. Let us denote the foot of the altitudes from the vertices 
and 
to the opposite sides by 
and 
respectively, and the intersection point of the altitudes of the triangle 
by 
Let 
be the intersection of the line 
and the segment 
A straight line passing through and perpendicular to 
intersects 
at 
Let 
be the intersection of the segment 
with the perpendicular drawn from 
Prove that
is the midpoint of segment 
This post has been edited 1 time. Last edited by falantrng, Aug 1, 2023, 12:29 PM
Reason: Typo
Reason: Typo
.
