Creating and manipulating expressions and formulae

problem
Favourite

Iff

Age

14 to 18

Challenge level

Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?
problem
Favourite

Always Perfect

Age

14 to 18

Challenge level

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
problem
Favourite

Leonardo's Problem

Age

14 to 18

Challenge level

A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?
problem
Favourite

System Speak

Age

16 to 18

Challenge level

Five equations... five unknowns... can you solve the system?
problem
Favourite

Sums of Squares

Age

16 to 18

Challenge level

Can you prove that twice the sum of two squares always gives the sum of two squares?
problem

Absurdity Again

Age

16 to 18

Challenge level

What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?
problem

Old Nuts

Age

16 to 18

Challenge level

In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?
problem

Incircles

Age

16 to 18

Challenge level

The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?
problem

Three Ways

Age

16 to 18

Challenge level

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.
problem

Reciprocals

Age

16 to 18

Challenge level

Prove that the product of the sum of n positive numbers with the sum of their reciprocals is not less than n^2.