2003 Moldova National Olympiad

Part of Moldova National Olympiad

Subcontests

(7)

1

Moldova MO

\text{Prove that each integer}

n\ge3

can be written as a sum of some consecutive natural numbers only and only if it isn't a power of 2

1

Rational logarithm

Find all integers n for which number \log_{2n\minus{}1}(n^2\plus{}2) is rational.

1

Find prime numbers

Find all prime numbers

a,b,c

that fulfill the equality: (a\minus{}2)!\plus{}2b!\equal{}22c\minus{}1

1

Properties of derivatives

For every natural number

n\geq{2}

consider the following affirmation

P_n

: "Consider a polynomial

P(X)

n

) with real coefficients. If its derivative

P'(X)

n-1

distinct real roots, then there is a real number

C

such that the equation

P(x)=C

n

real,distinct roots." Are

P_4

P_5

both true? Justify your answer.

1

Limit involving the fibonacci sequence

(F_n)_{n\in{N^*}}

be the Fibonacci sequence defined by

F_1=1

F_2=1

F_{n+1}=F_n+F_{n-1}

n\geq{2}

. Find the limit:

\lim_{n \to \infty}(\sum_{i=1}^n{\frac{F_i}{2^i}})

1

Intricate sequence

For every natural number

n

a_n=ln(1+2e+4e^4+\dots+2ne^{n^2})

\displaystyle{\lim_{n \to \infty}\frac{a_n}{n^2}}

1

Special division of polynomials

Consider the polynomial

P(x)=X^{2n}-X^{2n-1}+\dots-x+1

n\in{N^*}

. Find the remainder of the division of polynomial

P(x^{2n+1})

P(x)