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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 is both strictly greater than the average of 's column and strictly less than the average of 's row.
be a positive integer . Prove that there exists infinitely many pairs of positive integers 
such that 
and :
and 
on a given circle meet at point 
. Let 
be a point on segment 
and let 
meet the circle again at 
. The line through 
parallel to 
intersects 
at 
. Prove that 
is parallel to if and only if 
.
and on a given circle meet at point . Let be a point on segment and let meet the circle again at . The line through parallel to intersects at . Prove that is parallel to if and only if .
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 
