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 scalene triangle with circumcenter 
. Let 
be the feet of the altitudes from 
to 
respectively. Let 
be the midpoint of 
. Let 
be the circle with diameter 
. Let 
be the intersection of 
and 
. Let 
be the orthocenter of . Let 
be the intersection of 
and . Let 
be the lines through 
tangent to 
respectively. Let 
be the intersection of 
and 
. Let 
be the intersection of 
and 
. Let 
be the line through parallel to 
and let 
be the reflection of across . Prove that 
is tangent to 
.
+1 w
be a positive integer, and let its positive divisors be![\[
d_1 < d_2 < \cdots < d_k.
\]](http://latex.artofproblemsolving.com/6/3/e/63ec82aba0d6c4eece89af6a1247a89498616281.png)
Define 
to be the number of ordered pairs 
with 
such that 
.
. when![\[
n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k},
\]](http://latex.artofproblemsolving.com/2/7/2/272dc2cc33ebbf695311cb61567fa0297e1299b6.png)
where the 
are distinct primes and the 
are positive integers. be the circumcenter of an acute-angled triangle with 
. The line 
meets the side at . The circumcenters of the triangles 
and 
are 
and 
, respectively. Extend the sides 
and 
beyond 
, and choose on the respective extensions points 
and such that 
and 
. Prove that the quadrilateral 
is a rectangle if and only if 
.
is a prime number
is a positive integer. Call all positive divisors of which are different from 
and beautiful divisors.We call a special number when it has at least 
beautiful divisors and difference of any beautiful divisors divides as well. Find all special numbers.
such that 
.
is defined by 
and for all positive integers 
, 
. Show that for every positive integer 
, there exists a suitable positive integer 
such that 
is a multiple of .
be a positive integer. Determine all sequences 
of positive integers such that
for all positive integers 
.
be an odd prime. Define 
over the positive integers as follows:
be chosen such that there exists an ordered pair of positive integers 
where 
such that 
. Prove that there exists at least three integers 
such that 
and 
is a perfect square.
be the nonnegative integers in 
where 
. Define 
. Prove that 
. such that 
where 
, and the statement is still true when 
? such that 
where 
, and the statement is still true when ?.
satisfying :![\[n^{2p}=m^2+n^2+p+1\]](http://latex.artofproblemsolving.com/7/4/e/74ee592cb8b3f3a3996ea34f40b78fde9d3672fc.png)
and are integers and is a prime number.
such that 
divides 
.
be a bijective map from a plane to itself such that:
is a line for every line 
,
is parallel to for every line .
be?
be a bijective map from a plane to itself such that: is a line for every line , is parallel to for every line . be?
and draw point 
.
which goes through point 
will go through 
and which goes 
is the intersection between the parallel and 
is a parallelogram we have that 
.
