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.
and 
be concentric circles, with in the interior of . From a point 
on one draws the tangent 
to (
). Let 
be the second point of intersection of and , and let 
be the midpoint of . A line passing through intersects at 
and 
in such a way that the perpendicular bisectors of 
and 
intersect at a point 
on . Find, with proof, the ratio 
.
not be a trapezoid such that there is a circle centered at 
that is tangent to the four sides 
, 
, 
, and 
. Let 
, 
, 
, and 
be the circumcenters of the triangles 
, 
, 
, and 
, respectively. Prove that there is a circle containing the circumcenters of the triangles 
, 
, 
, and 
.
and 
for all 
be real numbers such that 
Prove that
be non-negative real numbers such that 
Prove that
be an acute non-isosceles triangle with circumcircle 
, circumcenter 
and orthocenter 
. We draw a line perpendicular to 
through and a line perpendicular to 
through . Prove that the points of intersection of these lines with sides and 
lie on a circle, which is tangent to . be a triangle with incenter and intouch triangle 
. Let 
be the foot of the perpendicular from onto 
. Assume that 
, 
intersect the sides , in 
respectively. Finally, let the rays 
, 
meet the circumcircle of 
in 
, respectively. Prove that the tangent from to the incircle and the line 
meet on the circumcircle of .![$[0, n]^2$](http://latex.artofproblemsolving.com/b/3/8/b38b78b68b1c5651fa5825bf88eb10f801f4d821.png)
in the plane, for natural 
. Erase all the points which have both coordinates non-integer. We are left with one-dimensional cell complex which we will call . Find the maximal 
such that to 
there is a point with at least 
preimages.
of the![$[0, n]^3\subset R^3$](http://latex.artofproblemsolving.com/8/1/c/81c72fc51fd486cfe352438f452965f887fbf8f9.png)
by erasing all the points with all coordinates non-integer.
such that there exists exactly different functions 
such that ![$f:[1,5]\rightarrow[1,5]$](http://latex.artofproblemsolving.com/6/2/0/620e1621905a4670627d46fcb367e1a8c461bb49.png)
satisfying 
.
kg sky diver that jumps out of a plane with no initial velocity and an air resistance of 
. For this example assume that the positive direction is downward.
kg; 
m/s^2, we obtain the IVP: 
.
![\[T_1 z = \frac{z + 2}{z + 3}, \qquad T_2 z = \frac z{z + 1},\]](http://latex.artofproblemsolving.com/b/7/2/b72b5e302cecdf2649c43a078aea3aa9a88ef2cc.png)
find 
and 
and 
in terms of 
and 
