3.6 Elementary functions

E79

[09G]Let \(n,m,k\) be positive integers. Prove that the number \( (n+\sqrt{m})^ k + (n-\sqrt{m})^ k \) is integer.

Hidden solution: [UNACCESSIBLE UUID ’09H’]

E79

[09J]Let \(K\) be a positive integer, \(N\) an integer, and \(I=\{ N,N+1,\ldots ,N+K\} \) be the sequence of integers from \(N\) to \(N+K\). For each \(n∈ I\) we set an integer values \(a_ n\). Let \(p\) be the only one polynomial of degree \(K\) such that \(p(n)=a_ n\) for every \(n∈ I\).

• Show that \(p\) has rational coefficients.

• Show that \(p(x)\) is integer for every \(x\) integer.

• Find an example of a polynomial \(p\) which takes integer values for \(x\) integer, but not all coefficients of \(p\) are integers.

• What happens if \(I\) contains \(K+1\) integers, but not consecutive? Is it still true that, defining \(p(x)\) as above, \(p\) only assumes integer values on integers?

E79

[09K]Let \(p(x)\) be a polynomial with real coefficients of degree \(n\), show that exists \(c{\gt}0\) such that for every \(x\) we have \(|p(x)|≤ c(1+|x|^ n)\). Hidden solution: [UNACCESSIBLE UUID ’09M’]

E79

[211](Proposed on 2022-12) Prove that, for \(n≥2\),

\[ ∑_{k=1}^{n-1} \frac 1{k} ≥ \log (n) \]

Hidden solution: [UNACCESSIBLE UUID ’212’]