How find the equality $\lambda$ of min

Question

let $n\ge3,n\in N$,and $\alpha,\beta,\gamma\in(0,1),a_{k},b_{k},c_{k}\ge 0,k=1,2,\cdots,n$ and such $$\sum_{k=1}^{n}(k+\alpha)a_{k}\le\alpha,\sum_{k=1}^{n}(k+\beta)b_{k}\le\beta,\sum_{k=1}^{n}(k+\gamma)c_{k}\le\gamma$$

if for any such above conditions $a_{k},b_{k},c_{k}(k=1,2,\cdots,n)$ then have $$\sum_{k=1}^{n}(k+\lambda)a_{k}b_{k}c_{k}\le\lambda$$ find the $\lambda_{\min}$

This Problem is from china South East Mathematical Olympiad of 2013,7,26,and I think This problem is not easy,so I hope someone can help,Thank you

Answer 1

Simply apply Lagrangian dual method, you will get the result.