satisfying 
,![\[ \frac{a^4+5b^4}{a(a+2b)} + \frac{b^4+5c^4}{b(b+2c)} + \frac{c^4+5a^4}{c(c+2a)} \geq 1- ab-bc-ca \]](http://latex.artofproblemsolving.com/f/e/8/fe85cf52b883c214806ab837a2eb25cebb511d7b.png)
such that there exists a unique integer 
such that 
with the following property:![\[ n!\mid a^n + 1
\]](http://latex.artofproblemsolving.com/e/2/3/e23d39aab685888993d355d8c93d4e6475f53487.png)
(use in a problem for junior) to practice. Can anyone share to me?
Y by
The acute triangle 
is inscribed in a circle with center 
. Let 
be the intersection of the bisector of angle 
with segment 
and 
the intersection point of 
with the perpendicular on 
passing through . Show that 
.
is inscribed in a circle with center 
. Let 
be the intersection of the bisector of angle 
with segment 
and 
the intersection point of 
with the perpendicular on 
passing through . Show that 
.This post has been edited 2 times. Last edited by parmenides51, Sep 27, 2021, 1:47 PM
be the intersection of line segments 
and 
.
be the foot of the perpendicular from 
to 
and 
are congruent.
are isogonal lines. Therefore, 
is the bisector line of angle 
.
and 
From 
and 
we get that triangles 
and 
are congruent, and the problem has been solved.
