[/quote]
,
and 
Prove that 
be a triangle. A circle passing through 
intersects segments 
and 
at 
and 
,
and 
intersect at 
,
and 
intersect at 
.
if and only if 
.
and 
.
be reals such that 
Prove that
Prove that
such that 
.
.
and 
as the reflections of 
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.
and 
be points on segment 
denote the circle passing through 
denote the circle passing through 
.
)
.
and 
,
[list=a]
and 
,
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
Let 
be a set of 
non-negative integers and 
be a function with the following two properties:
1) For every two distinct positive integers
there exists 
, such that 
divides 
.
2) For every positive integer
there exists a positive integer 
such that 
whenever ![$x,y \in [t, t+N]$](//latex.artofproblemsolving.com/9/e/4/9e4bbbdf20624022d4c0da2f8158b73ffbbc999d.png)
are distinct.
Prove that there are infinitely many primes
such that divides 
for some positive integer 
.
be a set of 
non-negative integers and 
be a function with the following two properties:1) For every two distinct positive integers
there exists 
, such that 
divides 
.2) For every positive integer
there exists a positive integer 
such that 
whenever ![$x,y \in [t, t+N]$](http://latex.artofproblemsolving.com/9/e/4/9e4bbbdf20624022d4c0da2f8158b73ffbbc999d.png)
are distinct.Prove that there are infinitely many primes
such that divides 
for some positive integer 
.
be the set of all primes dividing 
for 
.
such that![\[
v_p(D) > v_p(x + 1), \quad \forall x \in A.
\]](http://latex.artofproblemsolving.com/d/d/8/dd8b902d84abd0cf65e5dbec97792c1a1ee970f0.png)
By hypothesis, for every positive integer 
,
such that![\[
x - 1 \mid f(x + a) - f(1 + a).
\]](http://latex.artofproblemsolving.com/4/6/c/46cbf93d0627f9d11ee757e2873c9f602b432056.png)
If 
,![\[
v_p(f(x + a)) = v_p(f(1 + a)),\forall p\in S.
\]](http://latex.artofproblemsolving.com/1/6/5/165819128fddb28449d26115a339fd2f7cd180a2.png)
Since 
has no prime divisor outside ![\[
f(x + a) = f(1 + a).
\]](http://latex.artofproblemsolving.com/2/e/b/2eb56f97466ed46199e0d1dabceaf538e21407ad.png)
Thus, for all 
such that in every interval ![\( [t, t + N - \max A] \)](http://latex.artofproblemsolving.com/e/d/c/edcb5584c784a51d4d95a8b4b94d7550d8b196db.png)
,
,
.
such that![\[
f(x_i + a_i) = f(1 + a_i), \quad \forall i.
\]](http://latex.artofproblemsolving.com/d/4/c/d4c95cdd73e64709769675a0608df2f0e925de0b.png)
By the pigeonhole principle, there exist indices 
such that 
,![\[
f(x_i + a_i) = f(x_j + a_i).
\]](http://latex.artofproblemsolving.com/b/9/7/b976d306023e814c0d845d406d5ecde0c6ef933f.png)
This is absurd since 
and 
are distinct elements in ![\( [t, t + N] \)](http://latex.artofproblemsolving.com/1/a/c/1ac5e220d67774dfd0619f7fdc2e807021282a9a.png)
.
