[/quote]
,
be an integer. Find all sequences 
satisfying the following conditions:![\[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 + n;
\]](http://latex.artofproblemsolving.com/3/c/5/3c509ec2e9013e8d3be492c8eb44a7c33841b74e.png)
![\[ \text{ (b) } a_{i + 1} + a_{i + 2} + \ldots + a_{i + n} < a_{i + n + 1} + a_{i + n + 2} + \ldots + a_{i + 2n} \text{ for all } 0 \leq i \leq n^2 - n.
\]](http://latex.artofproblemsolving.com/9/7/d/97d2a467d1c0dc8594ec024c3bb9b8c87ee85b19.png)
be an acute triangle with circumcircle 
.
be the midpoint of 
and let 
be the midpoint of 
.
be the foot of the altitude from 
and let 
be the centroid of the triangle 
be a circle through 
.
and 
are collinear.
with integer coefficients such that ![$P\left(\sqrt[3]5+\sqrt[3]{25}\right)=5+\sqrt[3]5$](http://latex.artofproblemsolving.com/5/8/e/58ec2a921cd988bd5ae017a8653544ddd758225f.png)
.
-
.
,
in 
such that the sets ![\[ A_j=\{x+t_j\mid x\in A\},\qquad j=1,2,\ldots,100 \]](http://latex.artofproblemsolving.com/d/5/c/d5c2a7b943e8c7200361698f3eb6f83700ad68d6.png)
are pairwise disjoint.
such that for every index 
,
,
is a multiple of 
.
and 
be the points in which the median and the angle bisector, respectively, at 
.
and 
meets 
and 
,
the point in which the perpendicular at 
produced.
is perpendicular to 
and 
such that for all 
if 
then 
Find the smallest of sum of all elements of 
,![\[
\sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x
\]](http://latex.artofproblemsolving.com/9/5/5/955863a0cf8747ae45b736b0631f243d3908eb84.png)
such that there exist positive integers 
for which the following holds 
for all 
.
is a perfect cube
of a prime number 
and a positive integer 
for which 
is an integer.
is the centre of the excircle opposite the vertex 
This excircle is tangent to the side 
and 
,
and 
meet at 
,
and 
meet at 
Let 
and 
be the point of intersection of the lines 
and 
Prove that 
,
.
such that if 
and 
are distinct rational numbers satisfying 
or 
,
?
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
