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Ball Bearings

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

Air Routes

Find the distance of the shortest air route at an altitude of 6000 metres between London and Cape Town given the latitudes and longitudes. A simple application of scalar products of vectors.


The probability that a passenger books a flight and does not turn up is 0.05. For an aeroplane with 400 seats how many tickets can be sold so that only 1% of flights are over-booked?

Pdf Matcher

Age 16 to 18
Challenge Level


I have a series of six probability density functions $X$, each of which satisfies at least one of the following conditions:

  1. $X$ is non-negative.
  2. $X$ can take any positive value.
  3. $X$ is a reasonably realistic pdf for a model of a share price of a bank in a year's time.
  4. The probability of $X$ taking a value in the range $(a, b)$ is the same as the chance of $X$ taking a value in the range $(-b, -a)$.
  5. $X$ is a reasonably realistic pdf for a model of human life expectancy.
  6. There is a number $a$ such that $P(x> a) \geq 2 P(x> 2a)$.

Which of the following six curves (ignoring scale) would be the potential candidates for these mathematical descriptions? Are multiple matches possible? What axes and scales would you choose in each case?