[/quote]
,
have,where 
and 
inscribed in 
with 
being the midpoint of 
.
of 
.
be the projection of 
onto 
.
that is not on 
intersects 
at 
.
and 
intersect at 
.
is an isosceles triangle.
and let 
,
,
be a sequence of non-negative real numbers. Find the maximum value of 
in terms of 
.
as follows: write 
,
are non-negative integers and 
,
.
where 
and 
is the twist of 
for each positive integer 
.
if and only if the remainder when 
is either 
.
every next number (from the fifth number) is the unit number of the sum of the four numbers preceding it. Is there any cases that we get the numbers 
and 
in this series?
,
and 
meet at the point 
.
and 
.
prove that 
.
![$P \in \mathbb Q[x,y]$](http://latex.artofproblemsolving.com/a/3/8/a384cbf38fd3a7f812b22baaeb0860e7f6eb26c0.png)
with 
and ![$\forall (a,b) \in \mathbb Z^2 \cap [0,d]^2, P(a,b) \in \mathbb Z$](http://latex.artofproblemsolving.com/a/6/5/a65ecb4bb2d9b5e6096f322c2de8bd11b945ed6a.png)
.
?
.
be nonnegative real numbers such that 
.![\[
\frac{a}{4 - b} + \frac{b}{4 - c} + \frac{c}{4 - a} + \frac{1}{16}(1 - a)^2(1 - b)^2(1 - c)^2 \leq 1,
\]](http://latex.artofproblemsolving.com/5/b/d/5bd3349071e075519bd986c845c500125b7d46f8.png)
and determine all such triples 
where the equality holds.
such that 
is unbounded and![\[2f(m)f(n)-f(n-m)-1\]](http://latex.artofproblemsolving.com/0/f/7/0f77b5addbf8f5e603eb1b6923cdac795cd132e9.png)
is a perfect square for all integer 
as the number of orderings of n with badness less than
Y by GeoKing
In a class there are n students with unequal heights.
Find the number of orderings of the students such that the shortest person
is not at the front and the tallest person is not at the end.
Define the badness of an ordering as the maximum number 
such that there
are many people with height greater than in front of a person. For example:
the sequence
has badness 
since there are numbers greater
than
in front of it. Let 
denote the number of orderings of 
with badness . Find .
Find the number of orderings of the students such that the shortest personis not at the front and the tallest person is not at the end.
Define the badness of an ordering as the maximum number 
such that thereare
many people with height greater than in front of a person. For example:the sequence
has badness 
since there are numbers greaterthan
in front of it. Let 
denote the number of orderings of 
with badness . Find .This post has been edited 1 time. Last edited by Gloona, May 9, 2023, 3:49 PM
