be fixed points on this order on a line. Let 
be a variable circle through 
and 
and suppose it meets the perpendicular bisector of 
at the points 
and 
.
and 
be the other points of intersection of 
and 
with 
passes through a fixed point independent of 
be a convex quadrilateral with 
.
be the point of intersection of 
with 
and let 
be the midpoint of 
.
.
and 
be the projections of 
and 
.
meets again 
.
is the point of intersection of 
and 
intersect on the line 
.
.
be the circumcenter of this triangle 
,
,
,
,
,
the midpoints of these altitudes 
,
,
,
,
at the points 
,
,
,
and 
are concurrent. (When the point 
be an interior point for it such that 
.
When does equality hold?
Y by
Let 
be a positive integer. Alex plays on a row of 9 squares as follows. Initially, all squares are empty. In each turn, Alex must perform exactly one of the following moves:
Choose a number of the form 
, with 
a non-negative integer, and place it in an empty square.
Choose two (not necessarily consecutive) squares containing the same number, say . Replace the number in one of the squares with 
and erase the number in the other square.
At the end of the game, one square contains the number
, while the other squares are empty. Determine, as a function of , the maximum number of turns Alex can make.
be a positive integer. Alex plays on a row of 9 squares as follows. Initially, all squares are empty. In each turn, Alex must perform exactly one of the following moves:
Choose a number of the form 
, with 
a non-negative integer, and place it in an empty square.
Choose two (not necessarily consecutive) squares containing the same number, say . Replace the number in one of the squares with 
and erase the number in the other square.At the end of the game, one square contains the number
, while the other squares are empty. Determine, as a function of , the maximum number of turns Alex can make.This post has been edited 3 times. Last edited by EmersonSoriano, Apr 5, 2025, 5:26 PM
Reason: change
Reason: change
