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.
. Prove that for any given positive integer 
, the product
has at most distinct prime divisors.
such that:
for all 
be an acute-angled scalene triangle with circumcenter 
. Denote by 
, 
, and 
the midpoints of sides 
, 
, and 
, respectively. Let 
be the circle passing through 
and tangent to 
at . The circle intersects and 
at points 
and 
, respectively (where and are distinct from ). Let 
be the midpoint of segment 
, and let 
be the intersection of lines and 
. Prove that 
and that triangle 
is isosceles.
chessboard has a real number written in it in such a way that each number is equal to the geometric mean of all the numbers a knight's move away from it.
is inscribed in circle 
. The interior angle bisector of angle 
intersects side 
and at 
and 
(other than ), respectively. Let 
be the midpoint of side . The circumcircle of triangle 
intersects sides 
and 
again at 
and 
(other than ), respectively. Let 
be the midpoint of segment 
, and let 
be the foot of the perpendicular from to line 
. Prove that line 
is tangent to the circumcircle of triangle 
.
is written on the board. Anna, Boris, and Charlie can do the following actions: Anna can replace the number with its floor. Boris can replace any integer number with its factorial. Charlie can replace any nonnegative number with its square root. Prove that the three can work together to make any positive integer in finitely many steps.
and 
be different points on a circle such that 
is not a diameter. Let 
be the tangent line to at . Point 
is such that is the midpoint of the line segment 
. Point 
is chosen on the shorter arc of so that the circumcircle 
of triangle 
intersects at two distinct points. Let be the common point of and that is closer to . Line 
meets again at 
. Prove that the line 
is tangent to . be a right angled triangle with 
. Point is the midpoint of side And be an arbitrary point on 
. The reflection of 
over intersects lines and at and , respectively. The circumcircles of 
and intersect again at 
. Prove that the center of the circumcircle of 
lies on 
.
. The circle with diameter meets sides and at points D and 
, respectively. The tangents drawn to the circle at points and meet at point . Find the angle between lines 
and .
. The circle with diameter meets sides and at points D and , respectively. The tangents drawn to the circle at points and meet at point . Find the angle between lines and .
Prove that
.![\[\sqrt{a+\frac{9}{b+2c}}+\sqrt{b+\frac{9}{c+2a}}+\sqrt{c+\frac{9}{a+2b}}+\frac{2(ab+bc+ca)}{9}\ge\frac{20}{3}\]](http://latex.artofproblemsolving.com/b/3/2/b32e0bcba967ca847e878ac186c3168cd6c5fc66.png)
Prove that
Prove that
Where 
and 
Prove that
satisfying
Prove
and 
,
so 
.
is the antiparallel of 
so the angle between 