[/quote]
,
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 
.
![$a,b,c \in [-\frac{1}{2}, \frac{1}{2}]$](http://latex.artofproblemsolving.com/5/0/b/50bc691baa4e4bbef868e4799929c8c93d1b6f1a.png)
.
Let ![$a,b,c \in [-2,2]$](http://latex.artofproblemsolving.com/8/0/6/8066d4353091772c401e3e34f5499a8fa2bd9991.png)
.
be positive real numbers such that 
Prove that 
be prime numbers such that their sum isn't divisible by 
.
positive integer quadruples satisfy:
.
Prove that
that satisfy the equation 
.
![\[\sum_{k=1}^n\frac{\phi(k)}k\ln\frac{2^k}{2^k-1}\ge 1-\frac 1{2^n}.\]](http://latex.artofproblemsolving.com/3/d/3/3d32207efff8052214e65aca2505c9e4e7bb581c.png)
Y by
The quadrilateral 
is inscribed in the circle 
. The diagonals 
and 
meet at the point 
. Let 
and 
be the circumcircles of triangles 
and 
, respectively. On the arc 
, not containing the point , of the circle , the point 
is selected, and on the arc 
, not containing the point , of the circle , the point 
is selected so, that 
. The segment 
intersects at the points 
and 
. Prove that 
.
is inscribed in the circle 
. The diagonals 
and 
meet at the point 
. Let 
and 
be the circumcircles of triangles 
and 
, respectively. On the arc 
, not containing the point , of the circle , the point 
is selected, and on the arc 
, not containing the point , of the circle , the point 
is selected so, that 
. The segment 
intersects at the points 
and 
. Prove that 
.This post has been edited 1 time. Last edited by parmenides51, Dec 29, 2021, 2:56 PM
