[/quote]
,
be real numbers such that 
Prove that
such that 
.
.
and 
be the feet of the altitudes from 
and from 
to 
respectively.
and 
as the reflections of 
and 
be the circumcircle of 
.
the second intersection of line 
with 
the intersection of ray 
with 
is the circumcenter of 
can be inscribed within a circle.
of positive integers satisfying the following two conditions:
,
is in 
,
and 
;
such that 
is not divisible by 
.
?
then
where 
Also, 
and 
is largest real root of the equation
then 
Equality holds iff 
and any cyclic permutations.
such that the numbers 
si 
are simultaneously prime .
and 
be points on segment 
denote the circle passing through 
denote the circle passing through 
)
.
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 
.
Y by
Altitudes 
and 
are drawn in an acute-angled triangle 
. Points 
and 
are feet of perpendiculars drawn on the line 
from points 
and 
, respectively. What is the greatest length can have a segment 
, if 
?
and 
are drawn in an acute-angled triangle 
. Points 
and 
are feet of perpendiculars drawn on the line 
from points 
and 
, respectively. What is the greatest length can have a segment 
, if 
?This post has been edited 3 times. Last edited by parmenides51, Mar 10, 2021, 3:49 PM
and 
as the points on 
,
.
,
,
.
