[/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 the midpoint of 
and let 
be the midpoint of 
be the foot of the altitude from 
be the centroid of the triangle 
be a circle through 
.
and 
are collinear.
![$P\left(\sqrt[3]5+\sqrt[3]{25}\right)=5+\sqrt[3]5$](http://latex.artofproblemsolving.com/5/8/e/58ec2a921cd988bd5ae017a8653544ddd758225f.png)
.
-
.
,
in ![\[ A_j=\{x+t_j\mid x\in A\},\qquad j=1,2,\ldots,100 \]](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 
.
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 Adventure10, Mango247
Let 
be a triangle satisfying 
. Let 
be an arbitrary point on the side 
(different from 
and 
), and let the line 
meet the circumcircle of triangle at a point 
(apart from the point 
).
Let the circumcircle of triangle
meet the line 
at a point 
(apart from ), and let the circumcircle of triangle 
meet the line 
at a point 
(apart from ).
Prove that the excircle of triangle
at the side 
is identical with the excircle of triangle 
at the side 
if and only if the point is the midpoint of the arc on the circumcircle of triangle .
be a triangle satisfying 
. Let 
be an arbitrary point on the side 
(different from 
and 
), and let the line 
meet the circumcircle of triangle at a point 
(apart from the point 
).Let the circumcircle of triangle
meet the line 
at a point 
(apart from ), and let the circumcircle of triangle 
meet the line 
at a point 
(apart from ).Prove that the excircle of triangle
at the side 
is identical with the excircle of triangle 
at the side 
if and only if the point is the midpoint of the arc on the circumcircle of triangle .This post has been edited 2 times. Last edited by darij grinberg, Nov 5, 2005, 12:43 PM
are similar, and 
are, in fact, collinear.
-
.
of 
,
.
