[/quote]
,
![\[ \frac{\displaystyle\sum_{k=0}^r \binom nk \binom{n-2k}{r-k}}{\displaystyle\sum_{k=r}^n \binom nk \binom{2k}{2r}\left(\frac{3}{4}\right)^{n-k}\left(\frac{1}{2}\right)^{2k-2r}}\]](http://latex.artofproblemsolving.com/9/d/e/9dedd490eaf9c6d22aaf93bbd74d82375adcfa85.png)
\
.
is equal to
and 
for each positive integer 
,
?
of each imon 
.
become entangled if and only if the original imons 
are entangled, and each copy 
members from party 
members from party 
members from party 
members from party 
,
or more people involved).
that if they were to be considered the number of members from the four parties, it is possible that, after some time, all of the kidnapped people belong to the same party, under the same rules imposed by the wizard.
at 
meets sides 
and 
and 
respectively. Let 
be the circumcentre of 
is perpendicular to 
.
and ![$e\in [-2,2]$](http://latex.artofproblemsolving.com/b/b/7/bb78c9dedb18af5e6a904f5492917921e41a7ebc.png)
such that 
Find the value of 
be a positive integer such that 
,![\[
A = \{a_1, a_2, a_3, \ldots, a_m\}
\]](http://latex.artofproblemsolving.com/1/7/9/179687a134d5495ffff0b44fe763a2bb8b3ddd9c.png)
consist of distinct positive integers not exceeding 2025. Suppose that for every 
,
,
,
as well. Prove that![\[
\frac{a_1 + a_2 + a_3 + \cdots + a_m}{m} \geq 1013.
\]](http://latex.artofproblemsolving.com/c/0/d/c0d190f78d94deac1b947ba57a2de51d0b06f0bd.png)
Y by samrocksnature
In the parallelogram 
extend the bisectors of angles 
and 
and intersect with extensions of sides PN and 
at points 
and 
, respectively. Prove that the bisector of the original angle 
of the the parallelogram is perpendicular to 
.
extend the bisectors of angles 
and 
and intersect with extensions of sides PN and 
at points 
and 
, respectively. Prove that the bisector of the original angle 
of the the parallelogram is perpendicular to 
.This post has been edited 2 times. Last edited by parmenides51, Apr 26, 2021, 9:14 AM
and 
