Algebra

Vieta's Formula

Vieta's Formula: Level 5 Challenges

         

If the roots of p(x)=x3+3x2+4x8p(x) = x^3 + 3x^2 + 4x - 8 are a\color{red}{a}, b\color{blue}{b} and c\color{purple}{c}, what is the value of

a2(1+a2)+b2(1+b2)+c2(1+c2)?\color{red}{a}^2 \big(1 + \color{red}{a}^2\big) + \color{blue}{b}^2 \big(1 + \color{blue}{b}^2\big) + \color{purple}{c}^2 \big(1 + \color{purple}{c}^2\big)?

If α,β,γ\alpha , \beta , \gamma are roots of the equation x3+3x+9=0,x^{3} + 3x + 9 = 0, find the value of α9+β9+γ9.\alpha^{9} + \beta^9 + \gamma^{9}.

Consider all pairs of non-zero integers (a,b) (a,b) such that the equation

(axb)2+(bxa)2=x ( ax-b)^2 + (bx-a)^2 = x

has at least one integer solution.

The sum of all (distinct) values of xx which satisfy the above condition can be written as mn \frac{ m}{n} , where mm and nn are coprime positive integers. What is the value of m+nm + n ?

Let x1,x2,,x2015x_1,x_2,\ldots,x_{2015} be the roots of the equation x2015+x2014+x2013++x2+x+1=0. x^{2015} + x^{2014} + x^{2013} + \ldots + x^2 + x + 1 =0. Evaluate

11x1+11x2++11x2015. \frac1{1-x_1} + \frac1{1-x_2} + \ldots + \frac1{1-x_{2015}}.

What is the largest integer n1000 n \leq 1000 , such that there exist 2 non-negative integers (a,b)(a, b) satisfying

n=a2+b2ab1? n = \frac{ a^2 + b^2 } { ab - 1 } ?

Hint: (a,b)=(0,0) (a,b) = (0,0) gives us 02+020×01=0 \frac{ 0^2 + 0^2 } { 0 \times 0 - 1 } = 0, so the answer is at least 0. 0 .

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