[/quote]
,
and 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
weights, each weighing 
and another 
weights with 
each. Write a algorithm that determines the ways in which a scale can be balanced with a weight 
on the left pan, and display the number of possible solutions. (The weights can be placed on both pans and the program starts with the numbers 
.
![\[ x + y^2 = p^m \]](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 
,
,
and 
lie on a circle 
such that 
.
at 
and 
.
intersect again at 
.
.
with orthocenter 
.
to meet 
at 
meets the midpoint of 
,
.
Equality holds when 
intersect its circumcircle 
at 
.
determine a triangle 
,
is relative to 
are concurrent at a point 
and that 
be a cyclic quadrilateral whose opposite sides are not parallel. Suppose points 
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
Given a cube 
. Points 
are selected on edges 
and 
, and points 
are selected on edges 
and 
, respectively. The distance between straight lines 
and 
is equal to 
, the distance between straight lines 
and 
is equal to 
, the distance between straight lines 
and 
is equal to 
. It is known that the numbers 
are pairwise distinct. Prove that lines 
and 
meet at one point.
(folklore)
. Points 
are selected on edges 
and 
, and points 
are selected on edges 
and 
, respectively. The distance between straight lines 
and 
is equal to 
, the distance between straight lines 
and 
is equal to 
, the distance between straight lines 
and 
is equal to 
. It is known that the numbers 
are pairwise distinct. Prove that lines 
and 
meet at one point.(folklore)
This post has been edited 3 times. Last edited by parmenides51, Apr 12, 2021, 9:53 AM
