These are some STEP problems which can be stated very briefly and without technicalities, yet are very different from A level maths. Useful as “tasters”.

1. How many integers greater than or equal to zero and less than a million are not divisible by 2 or 5? What is the average value of these integers? (I/1999/1 second part)

2. The Fibonacci numbers F(n) are defined by the conditions F(0) = 0, F(1) = 1 and F(n+1) = F(n) + F(n−1) for all n ≥ 1. Compute F(n+1)F(n−1)−F(n)^{2} for a few values of n; guess a general formula and prove it. (II/1996/3 first part, abridged)

3. A small goat is tethered by a rope to a point at ground level on a side of a square barn which stands in a large horizontal field of grass. The sides of the barn are of length 2a and the rope is of length 4a. Let A be the area of the grass that the goat can graze. Prove that A ≤ 14πa^{2} and determine the minimum value of A. (I/2006/2)

4. 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 √6−√2 cm. (I/2001/1)

5. Bar magnets are placed randomly end-to-end in a straight line. If adjacent magnets have ends of opposite polarities facing each other, they join together to form a single unit. If they have ends of the same polarity facing each other, they stand apart. Find the expected number of separate units in terms of the total number N of magnets. (I/1999/13, first part)

6. A regular octahedron is a polyhedron with eight faces each of which is an equilateral triangle. Show that the angle between any two faces of a regular octahedron is arccos (−1/3). (I/2007/5, first part)

**Follow-up**

1. How many integers greater than or equal to zero and less than 4179 are not divisible by 3 or 7? What is the average value of these integers? (I/1999/1 second part)

2. By induction on k, or otherwise, show that F(n+k) = F(k)F(n+1) + F(k−1)F(n) for all positive integers n and k [if F(.) are the Fibonacci numbers] (II/1996/3, second part)

3. In the bar-magnets problem above, find the variance of the number of separate units in terms of the total number N of magnets. (I/1999/13, second part)

4. Find the ratio of the volume of a regular octahedron to the volume of a cube whose vertices are the centres of the faces of the octahedron. (I/2007/5, second part)

5. I choose at random an integer in the range 10000 to 99999, all choices being equally likely. Given that my choice does not contain the digits 0, 6, 7, 8 or 9, show that the expected number of different digits in my choice is 3.3616. (I/2012/13)

6. A thin non-uniform bar AB of length 7d has centre of mass at a point G, where AG = 3d. A light inextensible string has one end attached to A and the other end attached to B. The string is hung over a smooth peg P and the bar hangs freely in equilibrium with B lower than A. Show that 3 sin ∠PAB = 4 sin ∠PBA. Given that cos ∠PBA = 4/5 and that ∠PAB is acute, find in terms of d the length of the string and show that the angle of inclination of the bar to the horizontal is arctan (1/7) (I/2011/11)