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Omni-MATH / Define the function f:R→R by f(x)= begin{cases}frac{1}{x^{2}+sqrt{x^{4}+2 x}} & text { if } x notin(-sqrt[3]{2}, 0] 0 & text { otherwise }end{cases} The sum of all real numbers x …

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problem

Define the function f:R→Rf: \mathbb{R} \rightarrow \mathbb{R} by $$f(x)= \begin{cases}\frac{1}{x^{2}+\sqrt{x^{4}+2 x}} & \text { if } x \notin(-\sqrt[3]{2}, 0] \\ 0 & \text { otherwise }\end{cases}$$ The sum of all real numbers xx for which f10(x)=1f^{10}(x)=1 can be written as a+bcd\frac{a+b \sqrt{c}}{d}, where a,b,c,da, b, c, d are integers, dd is positive, cc is square-free, and gcd(a,b,d)=1\operatorname{gcd}(a, b, d)=1. Find 1000a+100b+10c+d1000 a+100 b+10 c+d.
Plain-text mathematical notation (without MathML)
Define the function f:R→R by $$f(x)= \begin{cases}\frac{1}{x^{2}+\sqrt{x^{4}+2 x}} & \text { if } x \notin(-\sqrt[3]{2}, 0] \\ 0 & \text { otherwise }\end{cases}$$ The sum of all real numbers x for which f¹⁰(x)=1 can be written as (a+b√(c))/(d), where a,b,c,d are integers, d is positive, c is square-free, and gcd(a,b,d)=1. Find 1000a+100b+10c+d.
Original LaTeX notation
Define the function $f: \mathbb{R} \rightarrow \mathbb{R}$ by $$f(x)= \begin{cases}\frac{1}{x^{2}+\sqrt{x^{4}+2 x}} & \text { if } x \notin(-\sqrt[3]{2}, 0] \\ 0 & \text { otherwise }\end{cases}$$ The sum of all real numbers $x$ for which $f^{10}(x)=1$ can be written as $\frac{a+b \sqrt{c}}{d}$, where $a, b, c, d$ are integers, $d$ is positive, $c$ is square-free, and $\operatorname{gcd}(a, b, d)=1$. Find $1000 a+100 b+10 c+d$.

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task_id
f832270f-9c8a-55f2-a81f-a5345332ec4d
task_key
test--f832270f-9c8a-55f2-a81f-a5345332ec4d
task_revision_id
1
config / split
/ test

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