# STEP III/2015/5

In the following argument to show that √2 is irrational, give proofs appropriate for steps 3, 5 and 6.

1. Assume that √2 is rational.

2. Define the set S to be the set of positive integers with the following property:

n is in S if and only if n√2 is an integer.

3. Show that the set S contains at least one positive integer.

4. Define the integer k to be the smallest positive integer in S

5. Show that (√2-1)k is in S.

6. Show that steps 4 and 5 are contradictory and hence that √2 is irrational.

Prove that $2^\frac{1}{3}$ is rational if and only if $2^\frac{2}{3}$ is rational.

Use an argument similar to that of part (i) to prove that $2^\frac{1}{3}$ and $2^\frac{2}{3}$ are irrational.

# STEP I/2007/5: the regular octahedron Above is half a rectangular octahedron being constructed in the car park of the industrial estate where I work. It’s being constructed as an artwork, In the Eye of the Storm, by Tom Paine and the Structured Light group, and includes LED lights placed along all the lengths of wood which change colour and switch on and off in a way devised by Tom. It’ll be on show in five art festivals this year.

And here is the STEP question. When it was in the STEP exam in 2007, and when we offered it to the students on the 2018-9 Greenwich STEP course as a practice problem, few students got anywhere with it. Mya Titus, however, cracked the problem without any help.

STEP I/2007/5:

(i) Show that the angle between any two faces of a regular octahedron is arccos(−13)

(ii) 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. Continue reading