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.
, but skips any number containing digits 
, 
or 
. If the last number he writes is 
, how many numbers has Penrick written in total?
of the set 
, define its colorfulness as the greatest natural number 
such that:
, 
, if 
, then 
.
? Determine how many such permutations have maximal colorfulness.
, 
, with 
, 
. Prove that there exists 
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 and , the notation 
represents 
. )
. The segment 
measures 40 units. Let 
be the orthocenter of triangle and 
its circumcenter. Let 
be the foot of the altitude from 
and 
the foot of the altitude from 
. Additionally, point divides the segment such that 
. If the perpendicular bisector of segment 
passes through point , calculate the area of quadrilateral 
.
be real number such that 
Prove that
Let 
be real number . Prove that
be a triangle. The points 
and 
lie on the segments 
and 
respectively, such that the lines 
and 
intersect in a common point. Prove that it is possible to choose two of the triangles 
and 
whose inradii sum up to at least the inradius of the triangle .
and 
be two positive integer numbers such that 
.Prove that 
divides 
be a point interior to triangle (with 
). The lines 
, 
and 
meet again its circumcircle 
at 
, 
, respectively . The tangent line at 
to meets the line 
at 
. Show that from 
follows 
.
integers is 
. After adding 
to each of the first four and subtracting 
from each of the last seven, what is the new mean?
and 
Prove that
Prove that
Let 
Prove that
