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 triangle, and let 
and 
be points on segment 
symmetric with respect to the midpoint of . Let 
denote the circle passing through and tangent to line 
at 
. Similarly, let 
denote the circle passing through and tangent to line 
at 
. Let the circles and intersect again at point 
(
). Prove that 
.
and 
, let us consider the equation
[list=a]
and 
, find the least positive integer 
satisfying this equation. and , there exist infinitely many positive integers satisfying this equation.
denotes the greatest common divisor of positive integers 
and 
.)
events 
that are each continuous and last on a certain time interval. Each event has a weight 
However, one can only choose to attend activities that do not overlap with each other. The goal is to maximize the sum of weights of all activities attended. Prove or disprove that the following algorithm allows for an optimal selection:
consider 
the sum of 
over all 
such that 
and are not compatible.
are 
spheres centered at 
, with radius of 
and , which is remarked as 
and 
, respectively. Given 
points 
outside of (not including the surface of ).
as the number of sets 
such that the midpoint of 
lies entirely inside of .
. There after each term is derived from the previous term by substituting 
for and for 
Thus the first few terms are , , 
,
?
is called frameable if it is possible to draw a regular polygon with sides all whose vertices lie on these lines, and no line contains more than one vertex of the polygon.
are frameable.
is not frameable.
is frameable.
, determine all polynomials 
with integer coefficients such that 
divides 
for every positive integer .
with the smaller base are constructed the squares 
and 
. Prove that the perpendicular bisector of passes through the midpoint of 
.
such that
for all real numbers 
satisfying the equation
has circumcenter 
. The lines 
and 
intersect at 
, while 
and 
intersect at 
and 
and 
intersect at 
. Show that if the triangles and 
are similar(with vertices in that order), than is equilateral.
candidates and judges, where 
is an odd integer. Each candidate is evaluated by each judge as either pass or fail. Suppose that each pair of judges agrees on at most candidates. Prove that ![\[{\frac{k}{m}} \geq {\frac{n-1}{2n}}. \]](http://latex.artofproblemsolving.com/3/8/7/387c8824b28f6422306e3c25b34fccb0a841f4cd.png)
be a triangle. Let 
be the feet of the altitudes from 
respectively. Let 
be the projections of onto line 
. Show that 
.
have the property that 
is a perfect square for infinitely many positive integers 
?
be a triangle, 
its circumcircle, 
its incenter, and 
a tangent circle to the line 
at and to the side . Prove that the circles and are tangent.
be tangent to the side at point 
and denote 
as the center of arc not lying on arc . By trilium theorem 
. Consider inversion centered at with radius 
. Points 
are obviously fixed and inversion swaps line with circumcircle . Because is tangent to line 
at this is also a fixed object under inversion. Since it’s tangent to it’s also tangent to the circumcircle. QED
?
