### Cosines Rule

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

### DOTS Division

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

### Root to Poly

Find the polynomial p(x) with integer coefficients such that one solution of the equation p(x)=0 is $1+\sqrt 2+\sqrt 3$.

# Janine's Conjecture

##### Stage: 4 Challenge Level:

Julia of Downe House School gave the neatest solution to this problem by substituting 'x-1', 'x', 'x+1' for the three consecutive numbers and giving the following statement of Janine's conjecture:

( x - 1) x ( x + 1) + x = x 3 .

This is Julia's proof:

( x - 1) ( x + 1) = x 2 - 1
and
( x 2 - 1) x = x 3 - x .

Therefore ( x - 1) x ( x + 1) + x = x 3 .

So Janine's conjecture will always work whichever three consecutive numbers are chosen.