satisfying
Prove
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
,![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](http://latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
,![\[\frac{2}{3}\left(\sqrt{a+b+c}+\sqrt{b+c+d}+\sqrt{c+d+a}+\sqrt{d+a+b}\right)\leq\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+d}+\sqrt{d+a}\leq 2(\sqrt{2}-1)\left(\sqrt{a+b+c}+\sqrt{b+c+d}+\sqrt{c+d+a}+\sqrt{d+a+b}\right).\]](http://latex.artofproblemsolving.com/2/d/4/2d48c32aff8f3b3992202f149d835e4ee0982c1b.png)
satisfying![\[f(x+y)f(x-y)=\left(f(x)+f(y)\right)^2-4x^2f(y),\]](http://latex.artofproblemsolving.com/c/6/2/c6237d13efcb3f2ef74cfdbede1a6583ce970e79.png)
For all 
.
Prove that
[/quote]
such that 
.
Where 
![$ a,b \in [0 ,1] . $](http://latex.artofproblemsolving.com/9/1/9/91980d5ca5d97fe4b61797e6fbb378211674d9cd.png)
Prove that
and 
Prove that 
be real numbers such that 
.
Prove that 
![$e\in [-2,2]$](http://latex.artofproblemsolving.com/b/b/7/bb78c9dedb18af5e6a904f5492917921e41a7ebc.png)
such that 
Find the value of 
:
be the altitude and 
be the intersection point of the altitudes of triangle 
is symmetric to 
.
intersects 
and 
.
.
where 
and 
where 
.
lie on the same circle. If 
and 
,
,
or 11. However, I didn't know how to find the radius. Can anyone give me a hint?
Y by donot, Adventure10, son7, Mango247
The triangle 
is isosceles with 
, and 
. The points 
and 
are chosen on the side 
such that, 
, and 
. Denote by 
the intersection point between the internal bisectors of the angles 
and 
. Compute 
.
is isosceles with 
, and 
. The points 
and 
are chosen on the side 
such that, 
, and 
. Denote by 
the intersection point between the internal bisectors of the angles 
and 
. Compute 
.
. By Angle chasing we find that 
.(1)
so
(2)
and 
.
then 
to mean the angle mod 
through which line 
must be rotated anticlockwise to become parallel with 
Substituting the first two into the sum of the second two gives 
,
.
.
.
)
)
,
.
.
