# STEP I 2001 Q.1 (triangle in square)

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 $\sqrt{6}-\sqrt{2}$ cm.

# Answer to STEP I 2013 Q.13

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

# Answer to STEP III 2007 Q.9

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
$mk^2a^2(\theta - \alpha)^2$
where k and α are constants satisfying k > 0 and $\frac{\pi}{4} < \alpha < \frac{\pi}{2}$
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