MathDB

3

Part of 2003 Hungary-Israel Binational

Problems(2)

integers between $n^2$ and $n^2 + n + 3\sqrt{n}$

Source: 16-th Hungary-Israel Binational Mathematical Competition 2003

4/1/2007
Let nn be a positive integer. Show that there exist three distinct integers between n2n^{2} and n2+n+3nn^{2}+n+3\sqrt{n}, such that one of them divides the product of the other two.
number theory unsolvednumber theory
set {s=\sum\frac{1}{x_i}|x_i\in\mathbb{N},s<d}

Source: 16-th Hungary-Israel Binational Mathematical Competition 2003

3/30/2007
Let d>0d > 0 be an arbitrary real number. Consider the set Sn(d)={s=1x1+1x2+...+1xn∣xi∈N,s<d}S_{n}(d)=\{s=\frac{1}{x_{1}}+\frac{1}{x_{2}}+...+\frac{1}{x_{n}}|x_{i}\in\mathbb{N},s<d\}. Prove that Sn(d)S_{n}(d) has a maximum element.
inductionnumber theory proposednumber theory