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then
where 
Also, 
and 
is largest real root of the equation
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.
If there are multiple such events, delete the event with the minimal weight.
are 
. Let internal and external angle bisector of angle 
intersect line in points 
and 
, respectively, and circumcircle of triangle 
intersects line in point 
. Prove that 
is angle bisector of 
. Let internal and external angle bisector of angle intersect line in points and , respectively, and circumcircle of triangle intersects line in point . Prove that is angle bisector of 
then 
Equality holds iff 
and any cyclic permutations.
is the Appolonius circle so 