[/quote]
,
and 
,
as the greatest exponent 
such that 
divides 
.
and infinitely many 
such that 
.
and 
.
with circumcircle 
is given. Let 
be the midpoint of 
,
be the foot of altitude from 
to 
,
be the intersection point of 
and 
.
lies on the arc 
.
.
be a trapezium inscribed in a circle 
with diameter 
. The circle with center 
and radius 
meets 
and 
(where 
)
at 
.
is perpendicular to 
.
![\[ x + y^2 = p^m \]](http://latex.artofproblemsolving.com/e/c/f/ecf553036f22327dd457dc06611b6bc1b0ef072c.png)
![\[ x^2 + y = p^n \]](http://latex.artofproblemsolving.com/5/b/d/5bd672934a395aabd6865ca4b50d8da75c0fe07e.png)
For 
positive integers and 
being a prime.
and 
.
Equality holds when 
or 
.
at 
intersect again at 
.
Equality holds when 
intersect its circumcircle 
.
determine a triangle 
,
is relative to 
are concurrent at a point 
and that 
lie in the interiors of segments 
respectively, such that 
Let 
intersect 
at 
,
intersect 
at 
.
and 
are either parallel or coincide.
and 
.
Where 
such that :
.
Y by Adventure10
We have a pool table 
meters long and 
meters wide with a single ball in the center. We throw the ball in a straight line and, after traveling 
meters, it stops at a corner of the table. How many times did the ball hit the edges of the table?
Note: When the ball rebounds on the edge of the table, the two angles that form its trajectory with the edge of the table are the same.
meters long and 
meters wide with a single ball in the center. We throw the ball in a straight line and, after traveling 
meters, it stops at a corner of the table. How many times did the ball hit the edges of the table?Note: When the ball rebounds on the edge of the table, the two angles that form its trajectory with the edge of the table are the same.
This post has been edited 1 time. Last edited by parmenides51, Sep 19, 2022, 10:05 PM
