Can you guess it?

Algebra Level 5

If the three real roots of the equation 7x3+7x27x+1=07x^3+7x^2-7x+1=0 are:

x1=aasec(mπb)x2=aa+sec(nπb)x3=aa+sec(pπb) \begin{aligned} x_1 &=&\dfrac{a}{a-\sec\left(\dfrac{m \pi}{b}\right)} \\ x_2 &=&\dfrac{a}{a+\sec\left(\dfrac{n \pi}{b}\right)} \\ x_3&=&\dfrac{a}{a+\sec\left(\dfrac{p \pi}{b}\right)} \\ \end{aligned}

Where x1<x2<x3x_1 < x_2 < x_3; a,b,m,n,pa, b, m, n, p are positive integers with gcd(m,b)=gcd(n,b)=gcd(p,b)=1 \text{gcd}(m,b)=\text{gcd}(n,b)=\text{gcd}(p,b)=1 and 2m,2n,2pb2m,2n,2p \leq b

Find 2b(a+m+n+p)2b-(a+m+n+p).

Inspired by this and this.

×

Problem Loading...

Note Loading...

Set Loading...