STEP I/2001/1 – The points A, B, and C lie on the sides of a square of side 1 cm and no two points lie on the same side. Show that the length of at least one side of the triangle ABC must be less than or equal to cm.
From the integers 1, 2, … 52, I choose seven (distinct) integers at random, all choices being equally likely. From these seven, I discard any pair that sum to 53. Let X be the random variable the value of which is the number of discarded pairs. Find the probability distribution of X and show that E(X) = 7 / 17 Continue reading
Mechanics of a wheel falling off the edge of a table. Continue reading
III/2007/9 – Two small beads, A and B, each of mass m, are threaded on a smooth horizontal circular hoop of radius a and centre O. The angle θ is the acute angle determined by 2 θ = AOB.
The beads are connected by a light straight spring. The energy stored in the spring is
where k and α are constants satisfying k > 0 and
The spring is held in compression with θ = β and then released. Find the period of oscillations in the two cases that arise according to the value of θ and state the value of β for which oscillations do not occur. Continue reading