Summation Summation Summation

Algebra Level 4

Given xk,yk,zk>0x_k, y_k, z_k > 0 for all kk satisfy the three summations:

k=12015xk=50k=12015yk=169k=12015zk=961 \begin{aligned} \displaystyle \sum_{k=1}^{2015} x_k & = & 50 \\ \displaystyle \sum_{k=1}^{2015} y_k & = & 169 \\ \displaystyle \sum_{k=1}^{2015} z_k & = & 961 \\ \end{aligned}

What is the minimum value of k=12015(xkykzk) \displaystyle \sum_{k=1}^{2015} \left ( x_k y_k z_k \right ) ?

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