Root Approximation: Level 3 Challenges Practice Problems Online

$\large \color{#D61F06}{3}x^{5} \color{#3D99F6}{=} \color{#D61F06}{1}x^{4} + \color{#D61F06}{4}x^{3} + \color{#D61F06}{1}x^{2} + \color{#D61F06}{5}x + \color{#D61F06}{9}$

The above polynomial has exactly one real root $\alpha$ . Find $\lfloor 1000 \times \alpha \rfloor$ .

Notation: $\lfloor x \rfloor$ denotes the greatest integer smaller than or equal to $x$ . This is known as the greatest integer function.

Consider this equation

$2^x=x^2$

There are 2 trivial solutions, namely $x=2$ and $x=4$

However, the equation is known to also have a negative solution. Find the negative solution.

Give your answer to 3 decimal places.

Find the value of $a$ to 3 significant figures, so that $x$ and $y$ are distinct real solutions to

$x+y=100\pi, \\ x^y = y^x = a.$

$\large x^y = y^x = 43$

If $x$ and $y$ are distinct positive real numbers satisfying the equation above, find $\lfloor 100(x+y) \rfloor$ .

You are allowed to use computer assistance.

You might refer to Soumava's algorithm.

$f(x) = 9x^8 + 7x^6 + 5x^4 + 3x^2 - 1 = 0$

Which option is closest to the largest real root of the equation above?

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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.

Root Approximation

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Root Approximation: Level 3 Challenges

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

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