3

Part of 1995 Vietnam Team Selection Test

Problems(2)

(a^3 + b^3)^n = 4*(ab)^1995

Source: Vietnam TST 1995, Problem 3

Find all integers

a

b

n

1

which satisfy \left(a^3 \plus{} b^3\right)^n \equal{} 4(ab)^{1995}

number theory unsolvednumber theory

n is periodic with the smallest period 1995

Source: Vietnam TST 1995, Problem 6

Consider the function f(x) \equal{} \frac {2x^3 \minus{} 3}{3x^2 \minus{} 1}.

1.

Prove that there is a continuous function

g(x)

\mathbb{R}

satisfying f(g(x)) \equal{} x and

g(x) > x

x

2.

Show that there exists a real number

a > 1

such that the sequence

\{a_n\}

, n \equal{} 1, 2, \ldots, defined as follows a_0 \equal{} a, a_{n \plus{} 1} \equal{} f(a_n),

\forall n\in\mathbb{N}

is periodic with the smallest period

1995

functionalgebra unsolvedalgebra