Problem 4

Let

x_1,x_2,\ldots,x_{2023}

be pairwise different positive real numbers such that

a_n=(x_1+x_2+\cdots+x_n)\left(\frac1{x_1}+\frac1{x_2}+\cdots+\frac1{x_n}\right)

is an integer for every

n=1,2,\ldots,2023

. Prove that

a_{2023}\ge3034

Step 1 of 3: The base case

In plain words

With only one term, the sum and the sum of reciprocals are simply x_1 and 1/x_1, whose product is always exactly 1.

a_1=1

Detailed analysis

For $n=1$ , $a_1=x_1\cdot\frac1{x_1}=1$ , which is indeed an integer, giving the starting value for the induction.