2

Part of 1985 Vietnam National Olympiad

Problems(2)

Find all functions

Source: Vietnam NMO 1985 Problem 2

Find all functions

f \colon \mathbb{Z} \mapsto \mathbb{R}

which satisfy: i) f(x)f(y) \equal{} f(x \plus{} y) \plus{} f(x \minus{} y) for all integers

x

y

f(0) \neq 0

iii) f(1) \equal{} \frac {5}{2}

functionalgebra unsolvedalgebra

Four solutions form an arithmetic progression

Source: Vietnam NMO 1985 Problem 5

Find all real values of parameter

a

for which the equation in

x

16x^4 \minus{} ax^3 \plus{} (2a \plus{} 17)x^2 \minus{} ax \plus{} 16 \equal{} 0 has four solutions which form an arithmetic progression.

parameterizationalgebra unsolvedalgebra