Show that there are infinitely many rational points on the unit circle and no rational points on the circle x^2+y^2=3.
Solving the equation x^3 = 3 is easy but what about solving equations with a 'staircase' of powers?
The symbol [ ] means 'the integer part of'. Can the numbers [2x]; 2[x]; [x + 1/2] + [x - 1/2] ever be equal? Can they ever take three different values?
Show that it is rare for a ratio of ratios to be rational.
Prove that sqrt2, sqrt3 and sqrt5 cannot be terms of ANY arithmetic progression.
Try this interactivity to familiarise yourself with the proof that the square root of 2 is irrational. Sort the steps of the proof into the correct order.
We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.
Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which could be the sides of a triangle.
Can you make a square from these triangles?
An introduction to proof by contradiction, a powerful method of mathematical proof.