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 Adventure10
In the triangle 
from the largest angle 
, 
let be the altitude. Lines 
and 
, are altitudes of triangles 
and 
respectively, 
and 
are bisectors of triangles 
and 
. Let 
be the foot of the perpendicular from the point 
on the line 
. Prove that is the intersection point of the angle bisectors of the triangle 
.
from the largest angle 
, 
let be the altitude. Lines 
and 
, are altitudes of triangles 
and 
respectively, 
and 
are bisectors of triangles 
and 
. Let 
be the foot of the perpendicular from the point 
on the line 
. Prove that is the intersection point of the angle bisectors of the triangle 
.This post has been edited 2 times. Last edited by parmenides51, Dec 15, 2022, 9:09 PM
,
and 
.
therefore 
,
is cyclic so 
.
bisects 
.
and done.
