[/quote]
,
that satisfy the equation 
.
be a circle, and let a line 
intersect 
and 
Circles 
and 
are internally tangent to 
and 
respectively, and both are tangent to 
Similarly, circles 
and 
are externally tangent to 
and 
respectively.
the segment 
has a length of 
and 
find the length of segment 
have the same incenter 
.
is concyclic iff 
is concurrent
be the nonnegative integers in 
where 
.
.
.
such that 
where 
,
?
where 
,
be a scalene triangle with circumcircle 
and circumcircle 
.
and 
as the midpoints of 
and 
,
to 
.
meets 
,
intersects 
.
at 
and 
at 
.
intersect 
,
intersect 
.
is the reflection of 
and 
lies on the circumcircle of 
.
be real numbers such that 
and 
and determine the equality cases.
be positive real integer such that 
Prove that
such that 
and 
satisfying :![\[n^{2p}=m^2+n^2+p+1\]](http://latex.artofproblemsolving.com/7/4/e/74ee592cb8b3f3a3996ea34f40b78fde9d3672fc.png)
and 
are integers and 
is a prime number.
and 
Prove that
Approach: I'm trying to factor 
to get 
.
such that 
divides 
.
Y by samrocksnature
Josée has a small cubic bottle of 
centimeters per side containing precious liquid that a merchant wants to buy from him. To fix the price, the merchant claims that the bottle is two-thirds full because if we place it so that a large diagonal of the cube (for example 
in the figure) to be perpendicular to the ground, there is liquid up to two-thirds of that diagonal. In order to find out if Josée has an interest in accepting this interpretation, find the fraction of the volume of the vial that the liquid actually represents.
centimeters per side containing precious liquid that a merchant wants to buy from him. To fix the price, the merchant claims that the bottle is two-thirds full because if we place it so that a large diagonal of the cube (for example 
in the figure) to be perpendicular to the ground, there is liquid up to two-thirds of that diagonal. In order to find out if Josée has an interest in accepting this interpretation, find the fraction of the volume of the vial that the liquid actually represents.This post has been edited 3 times. Last edited by parmenides51, Mar 13, 2021, 8:23 PM
of the diagonal is exactly where the surface of the water touches 3 of the vertices (if the ratio was in the interval 
,
.
