3

Part of 1990 Irish Math Olympiad

Problems(2)

Determine if f(x) exists

Source:

Determine whether there exists a function

f: \mathbb{N}\longrightarrow \mathbb{N}

such that f(n)\equal{}f(f(n\minus{}1))\plus{}f(f(n\plus{}1)) for all natural numbers

n\ge 2

functioninduction

Trigonometric Sequence

Source: 1990 IrMO Paper 2 Problem 3

t

be a real number, and let a_n=2\cos \left(\frac{t}{2^n}\right)-1, n=1,2,3,\dots Let

b_n

a_1a_2a_3\cdots a_n

. Find a formula for

b_n

that does not involve a product of

n

terms, and deduce that

\lim_{n\to \infty}b_n=\frac{2\cos t+1}{3}