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.
that satisfy 
be a sequence defined by 
. Then the product of four consecutive terms in can be written as the product of two terms in . Find 
if 
.
and a set 
consisting of 
disting positive integers smaller than 
are given. Prove that there exists a positive integer 
that can be written in the form 
, for 
in at least 
different ways.
with 
, 
and area 
. The circle 
with diameter 
meets 
at 
. Let the orthocenter of the triangle be 
, connect 
and extend to meet at 
such that 
and 
is the diameter of . Draw the circumcircle 
of 
, chord 
of is tangent to and it passes through , compute .
, point 
lies on side and point 
lies on side 
. Let 
and 
be points on segments 
and 
, respectively, such that 
is parallel to . Let 
be a point on line 
, such that lies strictly between and , and 
. Similarly, let 
be the point on line 
, such that lies strictly between and , and 
.
, and are concyclic.
has an incenter 
Suppose 
. Let 
be the midpoint of . Suppose that 
. 
meets 
again at point 
. Let points and be such that is the midpoint of 
and is the midpoint of 
. Point lies on the plane such that 
is a parallelogram, and suppose the angle bisectors of 
and 
concur on 
.
and 
meet at 
and 
. Prove that 
.
with another positive integer 
, as long as 
is a perfect square. For example, if the number on the blackboard is 
, Carmela can replace it with 
, because 
, then replace it with 
, because 
. If the number on the blackboard is initially , determine all integers that Carmela can write on the blackboard after finitely many operations.
with circumcenter 
and orthocenter , let 
be an arbitrary point on the circumcircle of triangle such that does not lie on line 
and that line 
is not parallel to line . Let be the point on the circumcircle of triangle such that 
is perpendicular to , and let 
be the point on line such that 
. Let and 
be the points on the circumcircle of triangle such that 
is a diameter, and 
and 
are parallel. Let be the midpoint of 
.
and 
are perpendicular.
and 
are perpendicular.
be positive integers and 
be an odd prime such that:![\[
a^p \equiv 1 \pmod{p^n}.
\]](http://latex.artofproblemsolving.com/3/4/8/34815b8211ded2f37b0653525dc4b436356e6eee.png)
Prove that:![\[
a \equiv 1 \pmod{p^{n-1}}.
\]](http://latex.artofproblemsolving.com/0/7/a/07a4ed2ab3c94f41028080f7789cfb6e2973955c.png)
be a cyclic quadrilateral. A circle centered at passes through 
and and meets lines 
and again at points and (distinct from 
). Let denote the orthocenter of triangle 
Prove that if lines 
are concurrent, then triangle and 
are similar.
be a convex trapezoid such that 
, 
, 
, and 
. Among all points inside the trapezoid satisfying 
, compute the minimum possible value of 
.
meet at point . lies on the circumcircle of 
. By similar triangle, 
. Assign coordinates, 
is the origin, 
. The minimum value of occurs when 
and the circumcenter of are collinear. Denote the center of 
as . lies on 
. Moreover, the perpendicular bisector of 
can be expressed as 
. Set it equal to 
. Thus, 
as desired.
such that 
satisfying 
and 
Then, note that
This implies that wherever 
must be cyclic. Since 
and 
is just the locus of points in the trapezoid that are on the circumcircle of 
and 
Then, it follows that 
after which it is not difficult to find that the circumcenter of 
is at 
Therefore, the circumradius of 
and the distance from the circumcenter of 
Hence, the minimum possible value of 
