2025 SSMO Team Round Problems/Problem 9

Problem

Pairwise distinct integers $a,$ $b,$ $c,$ and $d$ satisfy the system of equations $\begin{align*} ab &= cd \\ a+d &= b+c \\ \tfrac{1}{a} + \tfrac{1}{b} - \tfrac{1}{5} &= \tfrac{1}{c} + \tfrac{1}{d} + \tfrac{1}{5}. \end{align*}$ What is the minimum possible value of $a^2+b^2+c^2+d^2$ ?

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2025 SSMO Team Round Problems/Problem 9

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