[/quote]
,
such that the numbers 
si 
are simultaneously prime .
and 
,
[list=a]
and 
,
satisfying this equation.
denotes the greatest common divisor of positive integers 
and 
.
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 
If there are multiple such events, delete the event with the minimal weight.
spheres centered at 
,
and 
and 
,
points 
outside of 
as the number of sets 
such that the midpoint of 
lies entirely inside of 
.
for 
Thus the first few terms are 
,
.
is called frameable if it is possible to draw a regular polygon with 
are frameable.
is not frameable.
is frameable.
,
with integer coefficients such that 
divides 
for every positive integer 
and 
.
.
such that
for all real numbers 
satisfying the equation
has circumcenter 
.
and 
intersect at 
,
and 
intersect at 
and 
and 
intersect at 
.
are similar(with vertices in that order), than 
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 ![\[{\frac{k}{m}} \geq {\frac{n-1}{2n}}. \]](http://latex.artofproblemsolving.com/3/8/7/387c8824b28f6422306e3c25b34fccb0a841f4cd.png)
Y by
Let 
be a square. On the segments 
, 
, 
and 
, choose points 
and 
, respectively, such that 
. Let 
be the intersection point of 
and 
, 
be the intersection point of 
and , 
the intersection point of 
and , and 
the point of intersection of and . Prove that 
is a square.
be a square. On the segments 
, 
, 
and 
, choose points 
and 
, respectively, such that 
. Let 
be the intersection point of 
and 
, 
be the intersection point of 
and , 
the intersection point of 
and , and 
the point of intersection of and . Prove that 
is a square.
;
.
intersects the line 
in the point 
.
in the point 
.
intersects the line 
.
.
and the midpoint of 
is 
