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on each of six problems, and their total score is the sum of the six scores (out of 
). Given two students 
and 
, we write 
if there are at least five problems on which scored strictly higher than .
such that the following statement is true: for every integer 
, given students 
, \dots, 
satisfying 
, the total score of is always at most points more than the total score of .
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.
with 
, incircle 
, circumcircle 
Let 
be the midpoints of arcs 
, respectively. Prove that 
is tangent to 
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 
