2

Part of 1989 Vietnam National Olympiad

Problems(2)

Fibonacci sequence

Source: Vietnam NMO 1989 Problem 2

The Fibonacci sequence is defined by F_1 \equal{} F_2 \equal{} 1 and F_{n\plus{}1} \equal{} F_n \plus{}F_{n\minus{}1} for

n > 1

. Let f(x) \equal{} 1985x^2 \plus{} 1956x \plus{} 1960. Prove that there exist infinitely many natural numbers

n

f(F_n)

is divisible by

1989

. Does there exist

n

for which f(F_n) \plus{} 2 is divisible by

1989

number theory unsolvednumber theory

Sequence of polynomials

Source: Vietnam NMO 1989 Problem 5

The sequence of polynomials \left\{P_n(x)\right\}_{n\equal{}0}^{\plus{}\infty} is defined inductively by P_0(x) \equal{} 0 and P_{n\plus{}1}(x) \equal{} P_n(x)\plus{}\frac{x \minus{} P_n^2(x)}{2}. Prove that for any

x \in [0, 1]

and any natural number

n

it holds that 0\le\sqrt x\minus{} P_n(x)\le\frac{2}{n \plus{} 1}.

algebrapolynomialinequalities unsolvedinequalities