I am trying to compile a document of preparation questions to use in mock interviews with students applying to study Maths at top UK Universities.

I have many questions already, but it is always good to have more.

So what would be your favourite questions to ask students? These would have to be fairly short (whole question solvable in less than half an hour) with multiple stages or areas for further exploration/extension so that able candidates can get stretched.

Topics of interest are:

  • Number Theory Problems
  • Graph Sketching Problems
  • Complex Number Problems
  • Calculus Problems
  • Statistics
  • Mechanics Problems

Of course there are other fields, which I would appreciate your favourite questions about, and questions may even overlap, but I've put this list to hopefully get a broad coverage of the topics.

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    $\begingroup$ This question seems way too broad. $\endgroup$ – Joel Reyes Noche May 23 '20 at 14:29

Tom Bowler's booklet here has more than 100 questions like this.

I also think that the first part of some STEP I/II questions would be suitable for a mock interview.

A few other questions I am fond of:

  • Sketch $y=\dfrac{\ln x}{x}$ and hence determine which of $e^{\pi},\pi^e$ is larger.
  • Ignoring issues of convergence, work out the value of $\sqrt{2\sqrt{2\sqrt{2\sqrt{2\sqrt{2\ldots}}}}}$
  • $\begingroup$ Problem with that booklet is that many of them are very different to the actual interview questions at Cambridge for example, so students can get a bit of a shock in the real thing. My interview involved proving $e$ was irrational, some prime number question and other questions I can't really remember but know were a lot more complex than the existing compilations. I could do them but they were not what I was prepared for. All other people I know (sample > 10) were similar, but thank you nonetheless. $\endgroup$ – PhysicsMathsLove May 23 '20 at 15:18
  • $\begingroup$ Your own suggested questions are more similar, however. In fact I also now remember I had to work out $\sqrt{2 + \sqrt{2 + \sqrt{2 + \cdots}}}$ without thinking about convergence and then they of course asked me to think about convergence. $\endgroup$ – PhysicsMathsLove May 23 '20 at 15:23
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    $\begingroup$ I get the impression that admissions tests used by some Cambridge colleges, which then form the basis of the interview, have questions similar to STEP. There is definitely an old STEP question about the proof that $e$ is irrational. $\endgroup$ – A. Goodier May 23 '20 at 15:58

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