with 
.![\[
\frac{a^5}{b^2 + 2c^3} + \frac{2b^5}{3c + a^6} + \frac{c^7}{a + b^4} \geq 2.
\]](http://latex.artofproblemsolving.com/f/f/5/ff5043705025592aeefc55a764a8b7170e13ff4b.png)
satisfy 
.
and the incircle 
of 
at 
,
meets 
at 
and intersects 
at 
,
meets 
at 
.
(different from 
)
and 
.
.
,
,
,
.
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For 
represents 
.
Y by Adventure10, Mango247
The Fibonacci sequence is defined by 
and 
for 
. Let 
. Prove that there exist infinitely many natural numbers 
for which 
is divisible by 
. Does there exist for which 
is divisible by ?
and 
for 
. Let 
. Prove that there exist infinitely many natural numbers 
for which 
is divisible by 
. Does there exist for which 
is divisible by ?
