Vieta's Formula: Level 5 Challenges Practice Problems Online

If the roots of $p(x) = x^3 + 3x^2 + 4x - 8$ are $\color{#D61F06}{a}$ , $\color{#3D99F6}{b}$ and $\color{#69047E}{c}$ , what is the value of

$\color{#D61F06}{a}^2 \big(1 + \color{#D61F06}{a}^2\big) + \color{#3D99F6}{b}^2 \big(1 + \color{#3D99F6}{b}^2\big) + \color{#69047E}{c}^2 \big(1 + \color{#69047E}{c}^2\big)?$

If $\alpha , \beta , \gamma$ are roots of the equation $x^{3} + 3x + 9 = 0,$ find the value of $\alpha^{9} + \beta^9 + \gamma^{9}.$

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 $m + n$ ?

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

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

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 $0 .$

Concept Quizzes

Challenge Quizzes

Vieta's Formula: Level 5 Challenges

Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, interactive explorations.

Used and loved by over 7 million people

Learn from a vibrant community of students and enthusiasts, including olympiad champions, researchers, and professionals.