AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
. Does there exist a function $f: \mathbb{N}__0 \to \mathbb{N}_0$ such that:![\[ f^{2003}(n)=5n, \forall n \in \mathbb{N}_0 \]](http://latex.artofproblemsolving.com/7/a/d/7ad2ffcca35dbc32a84a35a86fefa260fe7c5a85.png)
and 
, 
?
and all positive integers 
and 
satisfying 
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
, where![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](http://latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
is interesting if 2018 divides 
(the number of positive divisors of ). Determine all positive integers 
such that there exists an infinite arithmetic progression with common difference whose terms are all interesting.
be an odd prime number and a sequence 
satisfy 
and 
Prove that always exists positive integer satisfying for all positive integers , then 
.
is "not congruence"
and 
Prove that
Solution:
and 
then 
and 
intersect at points 
and 
. A line through intersects and at points 
and 
, respectively. Line 
intersects at point 
, and line 
intersects at point 
. If 
is the circumcenter of 
, prove that 
.
, the incircle touches sides 
and 
at points 
and 
, respectively. Median of triangle from vertex meets segment 
at point 
. Prove that angle 
is obtuse.
be the tangency point of the incircle with . It is known that 
collinear. Let 
and 
intersect 
at 
respectively. It is known that the circle with diameter passes through . However, notice that is strictly in between points ; hence, is strictly inside the circle with diameter . So 
must be obtuse.
and 
.
,
.
that satisfies:
and 
.
![$$(\sqrt{x}+1)[2-(x-6)\sqrt{x-3}]=x+8$$](http://latex.artofproblemsolving.com/b/1/d/b1d283f2e7b8e574b6c0ff72236c52a39d3da9e8.png)
Prove that
Prove that
