AoPS Academy
be a random point on the smaller arc 
of the circumcircle of square 
, and let 
be the intersection point of segments 
and 
. The feet of the tangents from point 
to the circumcircle of the triangle 
are 
and 
, where 
is the center of the square. Prove that points 
, 
, and lie on a single circle.
and 
such that
such that for every 
is satisfied? Explain the answer.
, a cone 
has a vertex at 
, and its base is a circle lying on the plane
. The angle at the cone's vertex is 
. Write the equation of a plane 
that contains the x-axis and divides the cone into two parts with a volume ratio of 
(the part containing vertex 
can have the smaller volume).
and radius 
.
. and radius ..
where is a parallelogram.
. is connected to the midpoint of ![$[CT]$](http://latex.artofproblemsolving.com/f/3/f/f3f8e9913dc696e543e6a34d169ec3deb276c84d.png)
, point 
to the midpoint of ![$[DT]$](http://latex.artofproblemsolving.com/6/d/d/6dd49550ab42ec2f047eaf4b357ac4e7d1387b1e.png)
, to the midpoint of ![$[AT]$](http://latex.artofproblemsolving.com/6/e/3/6e397da3c2a38b457f7b3504c5cb2a488a64f3a6.png)
and point to the midpoint of ![$[BT]$](http://latex.artofproblemsolving.com/6/a/4/6a4b1bcea7921e762a359321ab861b6937f04782.png)
..
.
, is circumscribed around the balls. Then equals
with edges 
. three perpendiculars to the three lateral faces. from the apex such that these three points lie above the apex.
with edge 
cm.
and tangent to the plane 
. and a plane 
, parallel with the edges and 
.
.
and 
,
, above the xy-plane, such that the pyramid 
is a regular tetrahedron.
.
by the recurrence![\[
a(n) = \left| a(n-1) - a(n-2) \right|
\]](http://latex.artofproblemsolving.com/e/7/9/e7981725c3dfd9fd1a0e22c17eded1a4cde84e9f.png)
for all 
, with initial values 
, 
, where 
.
, there exists a positive integer 
such that![\[
a(i) = a(i+3)
\]](http://latex.artofproblemsolving.com/d/c/7/dc75144ca27873b2420e79b5e8f045be864a4c2e.png)
for all integers 
.
such that: 
