and 
and![\[P=a_1^2+a_2^2+...+a_{2024}^2-(a_1a_3+a_2a_4+...+a_{2022}a_{2024})\]](http://latex.artofproblemsolving.com/d/b/b/dbb88435dcbba1f5f43f99cdf1ee42aeea3f24ea.png)
Find maximum 
written on the board and then Beto erases them and writes one of the following numbers:
The game ends when only two numbers are left on the board and Ana cannot play. If the sum of the final numbers is a multiple of 3, Beto wins. Otherwise, Ana wins. ¿Who has a winning strategy?
be a fixed positive integer and consider an 
grid of real numbers. Determine the greatest possible number of cells 
in the grid such that the entry in 
be a positive integer . Prove that there exists infinitely many pairs of positive integers 
such that 
and :
Y by anantmudgal09, Adventure10
Around the triangle 
a circle is 
circumscribed , 
is the intersection point of the bisectors of this triangle. Line 
intersects for the second time at point 
. Let the circle with diameter 
intersect 
, 
and for the second time at points 
, 
and 
respectively. Lines 
and 
intersect at a point 
, and the segments 
and intersect at the point 
. Prove that 
.
a circle is 
circumscribed , 
is the intersection point of the bisectors of this triangle. Line 
intersects for the second time at point 
. Let the circle with diameter 
intersect 
, 
and for the second time at points 
, 
and 
respectively. Lines 
and 
intersect at a point 
, and the segments 
and intersect at the point 
. Prove that 
.
(since e.g. 
and 
)
is cyclic.![\[\angle IKA_1=\angle IKM=\angle IPM=90^\circ-\angle MIP=\angle CIA-90^\circ=\frac{\beta}{2}=\angle IBA=\angle IBA_1.\]](http://latex.artofproblemsolving.com/1/5/d/15d717df22bcb3eb3ab56026c23f1530cc0cc659.png)
Hence 
.
is cyclic and we conclude that 
.
as desired.
is isosceles.)
