[/quote]
,
and 
be points on segment 
symmetric with respect to the midpoint of 
denote the circle passing through 
at 
.
denote the circle passing through 
at 
.
)
.
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 
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 
with the smaller base 
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)
be the feet of the altitudes from 
respectively. Let 
be the projections of 
.
.
is a perfect square for infinitely many positive integers 
?
Y by GRCMIRACLES, Adventure10, Mango247
Let 
be an acute triangle. Let
and 
be points on ![$[AC]$](//latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
and ![$[BC]$](//latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
, respectively, such that 
and 
. Let 
be the circumcenter of 
, and 
be the circumcenter of 
, and 
be the circumcenter of 
. Find the ratio of area of 
and .
be an acute triangle. Let
and 
be points on ![$[AC]$](http://latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
and ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
, respectively, such that 
and 
. Let 
be the circumcenter of 
, and 
be the circumcenter of 
, and 
be the circumcenter of 
. Find the ratio of area of 
and .
and 
and 
respectively. Thus, 
and 
Also, 
and 
Extend 
until it intersects 
Then 
and 
are similar. If we denote by 
the heights from 
respectively, we get 
In addition, 
and 
Substituting 
in 
we get 
respectively. Then ![$[O_1O_2H]:[ABH]=1:4$](http://latex.artofproblemsolving.com/6/4/5/645280b9738466e189008f60bbb7e985b6621407.png)
.
.![$[O_1O_2H]=[O_1O_2O_3]$](http://latex.artofproblemsolving.com/3/c/1/3c138aaedf6c0fac2ba90a2a452278a9b29bf047.png)
.
.
