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, 2022

Number theory Difficulty 5.0 AIME, harder Prove it United Kingdom

(i) Expand and simplify $(x+1)(x^2-x+1)$ .

(ii) Find all powers of 2 that are one more than a cube.

(A power of 2 is a number that can be written in the form $2^n$ , where $n$ is an integer greater than or equal to 0.)

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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) added by this project. Solutions are the publisher's, linked not copied.