[/quote]
,
:
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)
such that 
that satisfy the equation
.
be points on the side BC such that 
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)
we have 
.
.
be a cyclic quadrilateral. The point 
the intersection of 
.
.
concyclic.
. Prove that: 
Y by Adventure10
Let 
be an acute triangle with 
. be 
the circumcircle circumscribed to the triangle and 
the midpoint of the smallest arc 
of this circle. Let 
and 
points of the segments 
and 
respectively such that 
. Let 
be the second intersection point of the circumcircle circumscribed to 
with . Let 
and 
be the intersections of lines 
and 
with other than 
, respectively. Let 
and 
be the intersection points of lines 
and 
with lines and respectively. Show that the 
line passes through the midpoint of
be an acute triangle with 
. be 
the circumcircle circumscribed to the triangle and 
the midpoint of the smallest arc 
of this circle. Let 
and 
points of the segments 
and 
respectively such that 
. Let 
be the second intersection point of the circumcircle circumscribed to 
with . Let 
and 
be the intersections of lines 
and 
with other than 
, respectively. Let 
and 
be the intersection points of lines 
and 
with lines and respectively. Show that the 
line passes through the midpoint of This post has been edited 1 time. Last edited by parmenides51, Jun 21, 2022, 1:39 AM
be the midpoint of 
.
,
.
.
are collinear, so we complete the proof.![[asy]/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
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be the midpoint of 
respectively.
it's easy to show that 
of 
since 
and 
is also diameter of 
implies that 
cyclic since 
so 
at 
and since we also have 
