Problem 1520

1. Given Conditions and Initial Setup:
We are given that $n \geq 3$ is an integer and $\alpha, \beta, \gamma \in (0,1)$. For every $a_k, b_k, c_k \geq 0$ (for $k = 1, 2, \dotsc, n$) with the constraints:
$$\sum_{k=1}^n (k+\alpha)a_k \leq \alpha, \quad \sum_{k=1}^n (k+\beta)b_k \leq \beta, \quad \sum_{k=1}^n (k+\gamma)c_k \leq \gamma,$$
we need to show that:
$$\sum_{k=1}^n (k+\lambda)a_k b_k c_k \leq \lambda.$$
Our goal is to find the minimum value of $\lambda$.

2. Initial Inequality Analysis:
Since $\sum abc \leq \sum (k+\alpha)a \sum (k+\beta)b \sum (k+\gamma)c \leq \alpha \beta \gamma < 1$, we can infer that:
$$\lambda \geq \frac{\sum k a b c}{1 - \sum a b c}.$$
We need to find the maximum of:
$$I = \frac{\sum k a b c}{1 - \sum a b c}.$$

3. Expression Simplification:
We express $I$ in terms of $\alpha, \beta, \gamma$:
$$I = \alpha \beta \gamma \frac{\sum k a b c}{\alpha \beta \gamma - \alpha \beta \gamma \sum a b c}.$$
Using the given constraints, we have:
$$I \leq \frac{\alpha \beta \gamma \sum k a b c}{\sum (k+\alpha)a \sum (k+\beta)b \sum (k+\gamma)c - \alpha \beta \gamma \sum a b c}.$$
This can be further simplified to:
$$I \leq \frac{\alpha \beta \gamma \sum k a b c}{\sum [(k+\alpha)(k+\beta)(k+\gamma) - \alpha \beta \gamma] a b c}.$$

4. Maximization and Final Bound:
Using the fact that $\frac{x+y}{a+b} \leq \max\left(\frac{x}{a}, \frac{y}{b}\right)$, we get:
$$I \leq \alpha \beta \gamma \max_{1 \leq k \leq n} \frac{k}{(k+\alpha)(k+\beta)(k+\gamma) - \alpha \beta \gamma}.$$
Therefore, we have:
$$I \leq \frac{\alpha \beta \gamma}{(1+\alpha)(1+\beta)(1+\gamma) - \alpha \beta \gamma}.$$

5. Attainability:
This bound is attainable by setting:
$$a_1 = \frac{\alpha}{1+\alpha}, \quad b_1 = \frac{\beta}{1+\beta}, \quad c_1 = \frac{\gamma}{1+\gamma},$$
and setting all other $a_k, b_k, c_k$ to zero.