2023 IMO Problems/Problem 4

Problem

Let $x_1, x_2, \cdots , x_{2023}$ be pairwise different positive real numbers such that $a_n = \sqrt{(x_1+x_2+···+x_n)(\frac1{x_1} + \frac1{x_2} +···+\frac1{x_n})}$ is an integer for every $n = 1,2,\cdots,2023$ . Prove that $a_{2023} \ge 3034$ .

Solution

We first solve for $a_1.$ $a_1 = \sqrt{(x_1)(\frac{1}{x_1})} = \sqrt{1} = 1.$ Now we solve for $a_{n+1}$ in terms of $a_n$ and $x.$

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2023 IMO Problems/Problem 4

Problem

Solution