Sign up to access problem solutions.

Already have an account? Log in here.

Vieta's formula relates the coefficients of polynomials to the sum and products of their roots. This can provide a shortcut to finding solutions in more complicated algebraic polynomials.

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

\[\color{red}{a}^2 \left(1 + \color{red}{a}^2\right) + \color{blue}{b}^2 \left(1 + \color{blue}{b}^2\right) + \color{purple}{c}^2 \left(1 + \color{purple}{c}^2\right)?\]

Sign up to access problem solutions.

Already have an account? Log in here.

Sign up to access problem solutions.

Already have an account? Log in here.

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

\[ ( ax-b)^2 + (bx-a)^2 = x \]

has at least one integer solution.

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

Sign up to access problem solutions.

Already have an account? Log in here.

Let \(x_1,x_2,\ldots,x_{2015} \) be the roots of the equation \[ x^{2015} + x^{2014} + x^{2013} + \ldots + x^2 + x + 1 =0.\] Evaluate

\[ \frac1{1-x_1} + \frac1{1-x_2} + \ldots + \frac1{1-x_{2015}}. \]

Sign up to access problem solutions.

Already have an account? Log in here.

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

\[ n = \frac{ a^2 + b^2 } { ab - 1 } ? \]

**Hint**:

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

Sign up to access problem solutions.

Already have an account? Log in here.

×

Problem Loading...

Note Loading...

Set Loading...