be an odd integer. Prove that there exists a prime number 
such that![\[
p \mid 2^{\varphi(n)} - 1 \quad \text{but} \quad p \nmid n.
\]](http://latex.artofproblemsolving.com/3/5/1/3514b64ecd5f7b2586d5bfeae59b7c9504d5b455.png)
be non-negative real numbers such that 
Prove that
be real numbers satisfying 
. Find the maximum value of![\[P = \dfrac{a(b+c)}{a^2+2bc+3} + \dfrac{b(a+c) }{b^2+2ca +3 } + \dfrac{c(a+b)}{c^2+2ab+3}.\]](http://latex.artofproblemsolving.com/7/b/2/7b2ec0a87ca05d3cb29da888a4dfa73e10f238f8.png)
and 
. Prove that 
Y by Adventure10, Mango247
In triangle 
, angles 
are acute. Point 
is on the side 
such that 
. Let the interior bisectors of 
meet 
at 
, respectively. If 
, prove that is isosceles.
, angles 
are acute. Point 
is on the side 
such that 
. Let the interior bisectors of 
meet 
at 
, respectively. If 
, prove that is isosceles.
then 
.![\[AB^2-BD^2=AC^2-CD^2
\]](http://latex.artofproblemsolving.com/f/3/d/f3d5d4e454f2527d0cefc70a0a01a2d5d43dea6b.png)
![\[\frac{AB^2}{BD^2}-1>\frac{AC^2}{CD^2}-1
\]](http://latex.artofproblemsolving.com/6/9/8/698a04d4271f5a9aac79103b5cd38bcc558fb5f2.png)
![\[\frac{AB}{BD}>\frac{AC}{CD}
\]](http://latex.artofproblemsolving.com/2/d/a/2da63681725f06ceadf262941172018cba391349.png)
![\[\frac{AE}{ED}>\frac{AF}{FD}\implies DF>DE
\]](http://latex.artofproblemsolving.com/3/5/c/35c676a4c6d03678d14f7bc540b2308affd10e32.png)
Thus we have 
,
.
.
.
.
.
.
=
=
.
.
and 
is an increasing function on 
)
.
and 
.
.
![$[0,\dfrac {\pi} {2}]$](http://latex.artofproblemsolving.com/1/a/8/1a8eca21213da2a7790690343dc41e1181722ba7.png)
)
.
.
