[/quote]
,
be non-negative real numbers such that 
Prove that
be a scalene triangle with 
,
be the foot of the altitude from 
.
be a point in the interior of the segment 
.
be the point on the segment 
such that 
.
be the point on the segment 
such that 
.
intersects segment 
at a second point 
(other than 
.
and 
.
for 
.
are integers, then 
is divisible by an odd prime.
![\[ \frac{\displaystyle\sum_{k=0}^r \binom nk \binom{n-2k}{r-k}}{\displaystyle\sum_{k=r}^n \binom nk \binom{2k}{2r}\left(\frac{3}{4}\right)^{n-k}\left(\frac{1}{2}\right)^{2k-2r}}\]](http://latex.artofproblemsolving.com/9/d/e/9dedd490eaf9c6d22aaf93bbd74d82375adcfa85.png)
\
.
is equal to
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 
.
is the square of an integer
Y by kiyoras_2001
Let 
be a positive integer. Given a sequence of nonnegative real numbers 
we define the transformed sequence 
as follows: the number 
is the greatest possible value of the average of consecutive terms of the sequence that contain 
. For example, the transformed sequence of 
is 
.
Prove that
a) For every positive real number
, the number of such that 
is less than or equal to 
.
b) The inequality
holds.
be a positive integer. Given a sequence of nonnegative real numbers 
we define the transformed sequence 
as follows: the number 
is the greatest possible value of the average of consecutive terms of the sequence that contain 
. For example, the transformed sequence of 
is 
.Prove that
a) For every positive real number
, the number of such that 
is less than or equal to 
.b) The inequality
holds.This post has been edited 1 time. Last edited by 407420, Jan 24, 2023, 3:30 PM
Reason: Fixed
Reason: Fixed
