[/quote]
,
Prove that
,
is rational for 
.
is called a rational point in 
-
.
and 
in 
and 
as 
.
,
,
be reals such that 
.Prove that
Where 
the equation 
and integers 
,
satisfying 
for 
(with the convention 
)![\[
\sum_{j=1}^n a_j^{2m-1} \leq \frac{mt}{2} \left( \sum_{j=1}^n a_j^{m-1} \right)^2.
\]](http://latex.artofproblemsolving.com/5/c/d/5cd52e9ce307563bde96f697a75079229d4bfcc2.png)
is inscribed in circle 
.
intersects side 
and 
and 
(other than 
be the midpoint of side 
intersects sides 
and 
again at 
and 
(other than 
be the midpoint of segment 
,
be the foot of the perpendicular from 
.
is tangent to the circumcircle of triangle 
.
prove that
and incircle 
.
and 
and 
respectively. Let 
and 
be points outside 
satisfying ![\[\angle BDX = \angle XEA = \angle YDC = \angle AFY = 45^{\circ}.\]](http://latex.artofproblemsolving.com/4/7/0/470b0486fa1342c4b6444ec1e3731c23bbb842fb.png)
Prove that the circumcircles of 
and 
.
such that 
be a fixed integer. The cells of an 
table are filled with the integers from 
to 
with each number appearing exactly once. Let 
of 
touch 
,
denote the line tangent to 
.
such that 
.
of 
.
and 
.![\[\sum_{i=1}^n (a_i-a_{i+1})^3\leq \frac{1}{2\sqrt{3}} (\sum_{i=1}^n a_i )^3,\]](http://latex.artofproblemsolving.com/8/1/0/810f07e3394370ce06aec1e7025bcf5cc5433d60.png)
where 
.
Y by
What is the least number required to colour the integers 
such that for any set of consecutive integers taken from the given set of integers, there will always be a colour colouring exactly one of them? That is, for all integers 
such that 
, there will be a colour coloring exactly one integer from the set 
.
such that for any set of consecutive integers taken from the given set of integers, there will always be a colour colouring exactly one of them? That is, for all integers 
such that 
, there will be a colour coloring exactly one integer from the set 
.This post has been edited 1 time. Last edited by kitun, Nov 15, 2021, 10:12 AM
