be the roots of 
with 
and 
.![$[|3 \alpha |] + [|6 \beta|] +
[|9 \gamma |]$](http://latex.artofproblemsolving.com/9/f/f/9ffb5118d94f0e2fb84a411b47ec91c122985673.png)
is equal to
, but no more than 
of primes (greater than 5) such that : 
such that 
where degrees are used.
and 
Prove that
Let 
Prove that
Let 
and 
Prove that
Let 
Prove that
were there in which 
beat 
,
,
and 
.
are all real positive number that: 
.
,
,
.
and 
.
Let 
.
Let 
.
Let 
.
Y by
Six points, labeled A, B, C, D, E, and F, are positioned consecutively on a straight line. Let G be a point not located on this line. The following distances are given: AC = 26, BD = 22, CE = 31, DF = 33, AF = 73, CG = 40, and DG = 30. Determine the area of triangle BGE.
I brute forced this with trig, was wondering if theres a more elegant way of doing this
I brute forced this with trig, was wondering if theres a more elegant way of doing this
,
,
,
,
.![\[
a + b = 26
\]](http://latex.artofproblemsolving.com/9/4/3/9431b176fc19d2ffff4afd0703406a16964d234e.png)
![\[
b + c = 22
\]](http://latex.artofproblemsolving.com/c/4/5/c45f2855e598b2f58e4fb5f37ced70e79296495e.png)
![\[
c + d = 31
\]](http://latex.artofproblemsolving.com/9/2/9/929150526293c50486d0ac2a3dfb974eaf9937e7.png)
![\[
d + e = 33
\]](http://latex.artofproblemsolving.com/1/b/e/1beeb3bb2c5e22bd29b4fd18aaa1695078762bf6.png)
![\[
a + b + c + d + e = 73
\]](http://latex.artofproblemsolving.com/b/2/3/b232d4a67c25ee65519b43c3067114ec7abb7599.png)
Solving this system yields:![\[
a = 18, \quad b = 8, \quad c = 14, \quad d = 17, \quad e = 16
\]](http://latex.artofproblemsolving.com/e/0/f/e0fe5b70e39bbec9aef037d22ceff6f21a25712e.png)
and 
,
is off the line.
,
has sides 
,
,
.
.
is computed using the formula:![\[
\frac{1}{2} \times \text{base} \times \text{height}
\]](http://latex.artofproblemsolving.com/5/c/1/5c1fc04b79ec70744041f0b0bee5fd5550372c7b.png)
where the base 
is:![\[
b + c + d = 8 + 14 + 17 = 39
\]](http://latex.artofproblemsolving.com/5/8/b/58bfa2efa58af85dd1de559f92db328d6b42e552.png)
Thus, the area of the triangle is:![\[
\frac{39 \times 24}{2} = 468
\]](http://latex.artofproblemsolving.com/8/2/8/82835a852911e1190580d2dcbb73e5a91284db20.png)
