[/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 an acute-angled triangle with 
.
be the circumcircle of 
be the midpoint of the arc 
of 
.
meets 
at 
and meets 
.
at 
.
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 Adventure10, Mango247
A positive interger 
is called rising if its decimal representation 
satisfies the condition 
. Polynomial 
with real coefficents is called interger-valued if for all integer numbers , 
takes interger values. is called rising-valued if for all rising numbers , takes integer values.
Does it necessarily mean that, "every rising-valued is also interger-valued "?
is called rising if its decimal representation 
satisfies the condition 
. Polynomial 
with real coefficents is called interger-valued if for all integer numbers , 
takes interger values. is called rising-valued if for all rising numbers , takes integer values.Does it necessarily mean that, "every rising-valued
is also interger-valued "?This post has been edited 1 time. Last edited by IstekOlympiadTeam, May 26, 2018, 11:00 AM
.
,
.
.
.
