Hard maths questions for higher maths GCSE

All of the questions that I've posted can be done with GCSE knowledge + insight. It's the insight that makes them hard, of course.

Circle A is inscribed inside a square of side 2 so that it is tangent to all four sides. A smaller circle, B, is inscribed in the gap between circle A and the two sides of the square at one of the corners. Circle B is tangent to circle A and to the two sides of the square.

Find the radius of circle B.

Solve $n^3-4n-315=0$

A computer password for a top secret system comprises 50 randomly chosen digits, each in the range 0-9. There are $10^{50}$ distinct such passwords.


A computer cracker builds a powerful computer to break into the system. It tries each password in turn until the correct one is found. It can try a million million million passwords per second, and it is left running for a million million million years.

To the nearest one millionth of one percent, what percentage of passwords are still left to be tried, after this period of time?

I would presume so, a small note however that the other roots are actually complex, not irrational.

My bad! I'll keep that in mind when doing something like this again...

It's not really an issue for GCSE level, I just thought I'd mention it for interest's sake, by definition an irrational number is by definition real.

It's not really an issue for GCSE level, I just thought I'd mention it for interest's sake, by definition an irrational number is by definition real.

If I'm being honest, I still don't get the difference between a complex number and an irrational number - do complex numbers only show up when working out quadratics?

To be fair, it's usually by convention that $n \in \mathbb{N}$ or $n \in \mathbb{Z}$ so being irrational would be a sufficient condition to discard those solutions, I think.

Essentially one defines an imaginary number $i=\sqrt{-1}$ and proceed from there, though that is a discussion best suited to another thread.

That's not what I got. I'll check my result - it came to a nice exact answer, with surds.

Yes but...
Hint:

I think that you misread the question. And that doesn't look quite right, even if you are trying to compute the % finished.

It would be easier to comment if you put up some working.

I did get an exact answer with surds, except I wasn’t sure howto put it down: (-1 + sqrt2)/2

That’s what I did at the beginning, except I didn’t know whereto go from n(n+2)(n-2) = 315, so I ended up using the factor theorem to get(n-7) and then used long division to get a quadratic.

I still haven’t familiarised myself with latex, so apologiesabout not being able to display everything in a way that’s easy to understand. Amillion million million passwords is the same as 10^17 passwords per second. Ithen multiplied this by 31536000, which is the number of seconds in a year, andthen multiplied by another 10^17 for the number of years. I divided all this by10^50 and multiplied by 100 to get a percentage. I’m not sure what else I could have done, unless I didmisread the question, or maybe made a stupid mistake.

This