[/quote]
,
:
'is a perfect square.
Prove that
be reals such that 
and 
Prove that
Where 
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of 
is the sum of 
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
and 
.
Let 
.
such that 
that satisfy the equation
.
at 
.
and 
be points on the side BC such that 
and 
,
and 
be the incenter of triangles 
and 
,
is the Euler line of triangle 
.
,
,
,
are odd primes and![\[2^ap^b=(p+2)^c-1.\]](http://latex.artofproblemsolving.com/5/8/5/58518a8ebed14836b72af743ff100eae2efba5e2.png)
Y by Instance
Points 
, 
and 
are taken on the sides of triangle 
, as shown in the figure. The angles marked equally turned out to be equal to each other: the angle 
is equal to the angle 
, the angle 
is equal to the angle 
, and the angle 
is equal to the angle 
. Prove that points , and are the midpoints of the sides of triangle .
, 
and 
are taken on the sides of triangle 
, as shown in the figure. The angles marked equally turned out to be equal to each other: the angle 
is equal to the angle 
, the angle 
is equal to the angle 
, and the angle 
is equal to the angle 
. Prove that points , and are the midpoints of the sides of triangle .This post has been edited 2 times. Last edited by parmenides51, Jun 6, 2021, 9:15 PM
and 
We can solve for other angles.
and 
are parallelograms. By using properties of parallelograms, we see that 
and 
Since 
and 
