[/quote]
,
blocks, each weighing at least 
.
with 
you can choose a subset of the blocks whose total weight is at least 
.
such that 
.
chessboard. Initially, Azer colors all cells of this chessboard with some colors. Then, Babek takes 
rows and 
cells in the intersection. Babek wants to have all these 
be non-negative real numbers such that 
Prove that
and 
Find the minimum 
be a positive integer and 
be real numbers such that ![\[a_1+a_2+\dots+a_n=0.\]](http://latex.artofproblemsolving.com/0/f/d/0fde8d75cebc35103ae9acf8ebe393653d336d60.png)
Define the set 
by![\[A=\left\{(i, j)\,|\,1 \leqslant i<j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\}\]](http://latex.artofproblemsolving.com/6/9/7/6974ebf20c44bcb4d52d85adc0805e28aafa5d6e.png)
Prove that, if ![\[\sum_{(i, j) \in A} a_{i} a_{j}<0.\]](http://latex.artofproblemsolving.com/b/e/3/be3e65a633dc2ac092a264968f1cf8514d3d6124.png)
is defined by 
and for all positive integers 
,
.
,
such that 
is a multiple of 
.
such that no sum of three distinct elements of 
.
.
has circumcircle 
and incenter 
.
at 
respectively. Circumcircle of triangle 
intersects 
distinct from 
,
intersects 
distinct from 
.
are concyclic.
.
is a parallelogram. Prove that 
.
Y by
There are 
segments, and it is known that the length of each is an integer in centimeters. The two shortest segments are 
centimeter, the longest is 
centimeters. Prove that among the segments you can choose three such that the length of one of they are less than the sum of the other two.
segments, and it is known that the length of each is an integer in centimeters. The two shortest segments are 
centimeter, the longest is 
centimeters. Prove that among the segments you can choose three such that the length of one of they are less than the sum of the other two.
