[/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 
Y by kiyoras_2001, Photaesthesia
Given a positive integer 
. A convex polygon is said to be 
-good if it contains lattice points where any three of them are not collinear.
(a) Show that there exists an-good convex polygon with area at most 
.
(b) Show that there exists a constant
so that any -good convex polygon has area at least 
.
Proposed by usjl
. A convex polygon is said to be 
-good if it contains lattice points where any three of them are not collinear.(a) Show that there exists an
-good convex polygon with area at most 
.(b) Show that there exists a constant
so that any -good convex polygon has area at least 
.Proposed by usjl
,
grid, such that no three of them lie on the same line by just taking 
for 
.
,
.
-
where the vertices are the lattice points that the polygon 
edges. Perhaps we can use hypergraph removal lemma to reach such a case.
