US-TST 2024 Problem 1

Find the smallest constant $C > 1$ such that the following statement holds: for every integer $n\geq 2$ and sequence of non-integer positive real numbers $a_{1}$ , $a_{2}$ , ..., $a_{n}$ satisfying

$$\frac{1}{a_{1}} +\frac{1}{a_{2}} +\dots +\frac{1}{a_{n}} = 1,$$

it's possible to choose positive integers $b_{i}$ such that

(i) for each $i = 1,2,\ldots ,n$ , either $b_{i} = \lfloor a_{i}\rfloor$ or $b_{i} = \lfloor a_{i}\rfloor +1$ ; and

(ii) we have

$$1< \frac{1}{b_{1}} +\frac{1}{b_{2}} +\dots +\frac{1}{b_{n}}\leq C.$$