Inequalities

problem
Eyes down

Age

16 to 18

Challenge level

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?
problem
In between

Age

16 to 18

Challenge level

Can you find the solution to this algebraic inequality?
problem
Farey neighbours

Age

16 to 18

Challenge level

Farey sequences are lists of fractions in ascending order of magnitude. Can you prove that in every Farey sequence there is a special relationship between Farey neighbours?
problem
Erratic quadratic

Age

16 to 18

Challenge level

Can you find a quadratic equation which passes close to these points?
problem
Making waves

Age

16 to 18

Challenge level

Which is larger cos(sin x) or sin(cos x) ? Does this depend on x ?
problem
Thousand words

Age

16 to 18

Challenge level

Here the diagram says it all. Can you find the diagram?
problem
Random inequalities

Age

16 to 18

Challenge level

Can you build a distribution with the maximum theoretical spread?