ABC is a triangle and D is a point on AB produced
beyond B such that BD = AC, and E is a point on AC produced
beyond C such that CE = AB. The right bisector of BC meets
DE at P. Prove that ÐBPC = ÐBAC.
The Fibonacci sequence { F_{n} } is defined by
F_{1} = F_{2} = 1 and F_{n+2} = F_{n+1} + F_{n} for
n = 0, ±1, ±2, ±3, ¼. The real number t
is the positive solution of the quadratic equation
x^{2} = x + 1.
(a) Prove that, for each positive integer n,
F_{-n} = (-1)^{n+1} F_{n}.
(b) Prove that, for each integer n, t^{n} = F_{n} t+ F_{n-1}.
(c) Let G_{n} be any one of the functions F_{n+1}F_{n},
F_{n+1}F_{n-1} and F_{n}^{2}. In each case, prove that
G_{n+3} + G_{n} = 2(G_{n+2} + G_{n+1}).
Prove that all functions P_{n} are polynomials.
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18.
Each point in the plane is coloured with one of three
distinct colours. Prove that there are two points that are unit
distant apart with the same colour.