2

Part of 2008 Federal Competition For Advanced Students, Part 2

Problems(2)

Polynomial with coefficients in integers

Source: Austrian Mathematical Olympiad 2008 Problem 2

(a) Does there exist a polynomial

P(x)

with coefficients in integers, such that P(d) \equal{} \frac{2008}{d} holds for all positive divisors of

2008

? (b) For which positive integers

n

does a polynomial

P(x)

with coefficients in integers exists, such that P(d) \equal{} \frac{n}{d} holds for all positive divisors of

n

algebrapolynomialVietanumber theory unsolvednumber theory

Find the missing positive integers

Source: Austrian Mathematical Olympiad 2008 Problem 5

Which positive integers are missing in the sequence

\left\{a_n\right\}

, with a_n \equal{} n \plus{} \left[\sqrt n\right] \plus{}\left[\sqrt [3]n\right] for all

n \ge 1

\left[x\right]

denotes the largest integer less than or equal to

x

g

with g \le x < g \plus{} 1.)

geometry3D geometrynumber theory unsolvednumber theory