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.
. 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
![\[f(n) = \sum_{k=0}^{n} \binom{n}{k} a_k, \quad n = 1, 2, 3, \ldots\]](http://latex.artofproblemsolving.com/9/3/7/9377324cb6585daaf2cbec7fed6fce2209835e22.png)
where![\[a_k =
\begin{cases}
3^k, & \text{if } k \text{ is even}, \\
0, & \text{if } k \text{ is odd}.
\end{cases}
\]](http://latex.artofproblemsolving.com/8/b/e/8be520a30bdfd9272163470ee69120c2fd3e9b59.png)
Find the remainder when 
is divided by 
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 
such that for all 
,![\[
mn + f(n!) = f(f(n))! + n \cdot \gcd(f(m), m!).
\]](http://latex.artofproblemsolving.com/c/0/c/c0cdf1bcb8053f5dc26e9df6355207b5e479f861.png)
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 
.
. Let 
be the midpoint of the arc 
not containing 
and define 
similarly. Show that the orthocenter of 
is the incenter 
of .
of positive integers such that 
is an integer.
table filled with natural numbers, we say it is a divisor table if:
-th row are exactly all the divisors of some natural number 
,
-th column are exactly all the divisors of some natural number 
,
for every 
.
is given. Determine the smallest natural number , divisible by , such that there exists an divisor table, or prove that such does not exist.
moves.
. Then suppose 
and 
correspond to the same state (some distinct values of 
and 
must satisfy this by PHP). Therefore, applying the algorithm 
times to a state returns that state back to itself. This is true no matter what state we are currently at, since every piece is unique on the cube. This means that after algorithms at the beginning, we will return to the solved state.
moves.
and then 
then trivially the puzzle gets resolved after applying this finite sequence of legal moves.
be positive reals such that 
,
of 
.
