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.
, such that for any 
, 
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 
. Prove that no very convex function exists.
satisfying
for all integers 
.
, and for any positive integer 
, 
means 
applied times to 
, and 
means 
applied times to .
+1 w
+1 w
satisfying 
.Prove that:
such that 
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 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 
. Let 
be the midpoint of 
. Denote by 
and 
the circumcircles of the triangles 
and 
, respectively. Let 
be the crossing point of the tangent to at 
with the tangent to at 
. Prove that 
is tangent to .
, let 
denote the feet of the altitudes from 
, respectively, and let be the circumcircle of 
. Let 
and 
be the circles through tangent to at and 
, respectively. Show that and meet at a point on 
other than .
such that![\[ f(x+f(y))=y+f(x+1),\]](http://latex.artofproblemsolving.com/b/a/3/ba35828951778c0d35f50b5b557796bd2cf9e338.png)
for all 
.
is defined as follows;
, and for each integer 
, 
is equal to the largest number that can be obtained by permutation of the digits of 
. Find the smallest for which the decimal representation of 
contains exactly 
digits
is defined as follows;, and for each integer , is equal to the largest number that can be obtained by permutation of the digits of . Find the smallest for which the decimal representation of contains exactly digits digits in the sequence 
one-digit numbers in the sequence : 
two-digits numbers in the sequence : 
, which is pretty number 
steps to go from to 
three-digits numbers in the sequence.
n-digits numbers where 
digits is 
digits is 
such that![\[
2^a \cdot 3^b + 1 = 5^c.
\]](http://latex.artofproblemsolving.com/0/a/c/0ac1f22162eacf698847bceb33f9a337bd53c44b.png)
and is in the form 
where 
are positive integers, 
is a nonnegative integer and 
is some integer in 
(with 
if 
)
as 
(this number contains 
decimal digits)
: pretty number 
: pretty number 
and 
: pretty number 
and we increased the number of digits
,
steps to go from 
to 
steps to reach 
steps to reach 
steps to go from 
to 
step to go from 
steps to go from 
to 
where 
steps to go from 
to 
)
steps to go from 
