Root Approximation

Root Approximation: Level 3 Challenges


3x5=1x4+4x3+1x2+5x+9 \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 α \alpha . Find 1000×α \lfloor 1000 \times \alpha \rfloor .

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

Consider this equation


There are 2 trivial solutions, namely x=2x=2 and x=4x=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 aa to 3 significant figures, so that xx and yy are distinct real solutions to

x+y=100π,xy=yx=a.x+y=100\pi, \\ x^y = y^x = a.

xy=yx=43 \large x^y = y^x = 43

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

You are allowed to use computer assistance.

You might refer to Soumava's algorithm.

f(x)=9x8+7x6+5x4+3x21=0 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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