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 a positive integer. Define a sequence by setting 
and, for each 
, letting 
be the unique integer in the range 
for which 
is divisible by 
. For instance, when 
the obtained sequence is 
. Prove that for any the sequence 
eventually becomes constant.
be an acute triangle with 
. The bisector of 
intersects the lines 
and 
at points 
and 
, respectively. The circumcircle of triangle 
has center 
and intersects the circumcircle of triangle at point 
with different to 
.
is tangent to the circumcircle of triangle .
satisfying
and 
such that 
is divisible by 
? be a triangle with incenter 
, and suppose the incircle is tangent to 
and at and . Denote by 
and 
the reflections of and over . Let 
be the intersection of with 
, and let 
be the midpoint of . Prove that 
and 
are perpendicular.
be a convex quadrilateral with perpendicular diagonals intersecting at 
. Prove that the reflections of across , BC, CD, DA$ are concyclic.
be three diameters of the circumcircle of an acute triangle . Let be an arbitrary point in the interior of 
, and let 
be the orthogonal projection of on 
, respectively. Let 
be the point such that is the midpoint of 
, let 
be the point such that is the midpoint of 
, and similarly let 
be the point such that is the midpoint of 
. Prove that triangle 
is similar to triangle .
such that for any 
, and for any real numbers 
there exists real numbers 
such that 
for at least 
indices 
. Here, 
and 
denotes the fractional part of 
.
be an acute-angled scalene triangle. Let 
and 
be points on the rays 
and 
, respectively, such that 
. Similarly, let 
and 
be points on the rays 
and 
, respectively, such that 
. Prove that if 
and 
intersect at 
, then 
is parallel to .
be the circumcircle of a triangle . Denote by and 
the midpoints of the sides and , respectively, and denote by 
the midpoint of the arc of not containing . The circumcircles of the triangles 
and 
intersect the perpendicular bisectors of and at points and , respectively; assume that and lie inside the triangle . The lines 
and 
intersect at 
. Prove that 
.
such that the largest prime divisor of 
is equal to the largest prime divisor of 
.
be the set of positive real numbers. Find all functions 
such that, for all positive real numbers , 
, 
and , the following three conditions are satisfied:
,
,
.
distinct divisors of a positive integer number 
form an arithmetic progression. Prove that the number has at least 
divisors.
for some 
. We may assume 
by replacing and 
by 
and 
respectively. Suppose 
for some 
, then 
we have 
, so 
. Since , 
, so for 
we have 
(
).
, plug in 
in) would imply 
, a contradiction. So 
and 
for some unique . Thus 
and 
are the only possible repetitions in the following 
divisors of : 
, which implies has at least 
divisors.
