[/quote]
,
,
,
,
are integers such that 
.
and 
of 
such that for each integer 
with 
,![\[ n\mid a_i - b_i - c_i
\]](http://latex.artofproblemsolving.com/1/a/e/1ae8d54f068b646ac7a1e43d75e7f26510cfdc95.png)
be an odd prime and 
be integers so that the integers 
are divisible by 
Prove that 
be real numbers such that 
Prove that 
Where 
be a triangle with circumcircle 
.
and 
respectively denote the midpoints of the arcs 
and 
that do not contain the third vertex. Let 
denote the midpoint of arc 
(the arc 
including 
)
be the incenter of 
be the circle that is tangent to 
be the circle that is tangent to 
,
friends, 
people are seated around a circular table in a fixed order. Ali places 
is the circumcenter of 
.
lie on 
,
.
is tangent to circumcircle of triangle 
.
be the sequence defined as follows: 
,
,
is the smallest integer greater than 
such that in 
there are no arithmetic progressions of length 
.
with 
,
be an integer. In a configuration of an 
board, each of the 
cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate 
counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of 
positive integers 
such that for every index 
,
,
is a multiple of 
.
Y by
Let 
be a rectangle and 
any point. Let 
be the foots of the perpendiculars from to the lines 
, respectively. Let the perpendicular bisectors of 
and 
intersect at 
, and those of 
and 
intersect at 
. Prove that the lines 
and 
are parallel.
be a rectangle and 
any point. Let 
be the foots of the perpendiculars from to the lines 
, respectively. Let the perpendicular bisectors of 
and 
intersect at 
, and those of 
and 
intersect at 
. Prove that the lines 
and 
are parallel.Attachments:
This post has been edited 4 times. Last edited by buratinogigle, Apr 17, 2025, 12:10 AM
Reason: Typo
Reason: Typo
