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Problem 658

AMC 10/12, early questions

Algebra Difficulty 3.0 Prove it UKMT Mathematical Olympiad for Girls · United Kingdom · 2014

(a) Find an integer solution of the equation $x^3 + 6x - 20 = 0$ and prove that the equation has no other real solutions. (b) Let $x$ be $\sqrt[3]{\sqrt{108} + 10} - \sqrt[3]{\sqrt{108} - 10}$ . Prove that $x$ is equal to $2$ .

This one wants a proof. Work it on paper, then check yourself against the publisher's own solution, linked below. Be honest about it: the record is only any use to you if it is.

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Source: UK Mathematics Trust, licensed © UK Mathematics Trust; past papers published free at ukmt.org.uk. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project. Solutions are the publisher's, linked not copied.