AoPS Academy
-tuple 
, 
, of symbols from 
, we define a sequence 
according to the following rule: If 
, then 
, where 
if 
(taking 
) and 
is the symbol other than 
if 
. Find all positive integers 
for which there exists some integer 
such that 
.
be an acute triangle with orthocenter 
. Let 
, 
and 
be the feet of the altitudes of triangle opposite to vertices 
, 
, and 
respectively. Let 
and 
be the midpoints of 
and 
, respectively. Let 
be the intersection of lines 
and 
. Prove that is the circumcenter of triangle if and only if is the midpoint of 
.
and 
be two prime numbers, with being a divisor of 
. Prove that there exist integers 
such that the polynomial 
is divisible by the polynomial 
with 
is a multiple of and 
.
lies inside triangle of sides of length 
. Show that if distances between and vertices of triangle are rational numbers then distances from to sides of triangle are rational numbers too.
, 
, 
. Let 
, 
, 
be the projections of into 
, 
and 
. Assume 1 or 2 or 3 of the distances from to sides are not rational, and use the fact that 
is rectangle.
and 
Prove that
