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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.

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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}.

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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: Challenge Level:2 Challenge Level:2

Janine noticed, while studying some cube numbers, that ``if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number cubed''; e.g., 3, 4, 5 gives 3 x 4 x 5 + 4 = 64, which is a perfect cube. Does this always work? Can you prove or disprove this conjecture?