Olympiad Problem 7

Algebra Level 5

{k1+k2++kn=5n41k1+1k2++1kn=1 \begin{cases} k_1 + k_2 + \cdots + k_n = 5n - 4 \\ \dfrac1{k_1} + \dfrac1{k_2} + \cdots + \dfrac1{k_n} = 1 \end{cases}

Let k1,k2,k3,,kn,nk_1, k_2, k_3, \ldots, k_n , n be all positive integers satisfying the system of equations above.

Find the sum of all distinct possible values of n+k1+k2++knn + k_1 + k_2 + \cdots + k_n .

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