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.
be an acute-angled triangle with 
. Let 
be the circumcircle of . Let 
be the midpoint of the arc 
of containing 
. The perpendicular from to 
meets 
at 
and meets again at 
. The line through parallel to meets line 
at 
. Denote the circumcircle of triangle 
by 
. Let meet again at 
. Prove that the line tangent to at 
meets line on the internal angle bisector of 
.
, 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
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 special if is the common two leading digits of the decimal expansion of 
and 
, where is a certain positive integer. Given that there are two special number, find these two special numbers.
, so if we let 
and 
, with 
, we must have 
where 
. Using the fact that 
, we must have 
. Therefore ![$s=\lfloor \sqrt[3]{10^{t+3}}\rfloor$](http://latex.artofproblemsolving.com/5/f/9/5f904f4877fc50d88c5acbf5edaab520ecd577da.png)
, where . If 
, we would have to have the equalities 
, which imply that and are powers of 
, which isn't possible since 
. We therefore conclude that the only possible values for are ![$s=\lfloor \sqrt[3]{10^{4}}\rfloor=21$](http://latex.artofproblemsolving.com/4/0/4/4045b409b7ffc8a9439d7aec300e3ae18bdcec08.png)
and ![$s=\lfloor \sqrt[3]{10^{5}}\rfloor=46$](http://latex.artofproblemsolving.com/4/3/6/436f3a767c7b2b3b95631d3ffacf80016f266d4a.png)
.
such that![\[
2^a \cdot 3^b + 1 = 5^c.
\]](http://latex.artofproblemsolving.com/0/a/c/0ac1f22162eacf698847bceb33f9a337bd53c44b.png)
