Tìm giá trị nhỏ nhất

To find the minimum value of

\[
P = \frac{\sqrt{y^2 + 2x^2}}{xy} + \frac{\sqrt{z^2 + 2y^2}}{zy} + \frac{\sqrt{x^2 + 2z^2}}{xz}
\]

given the constraint

\[
\frac{1}{x+y} + \frac{1}{x+z} + \frac{1}{y+z} \geq 2020,
\]

we can approach the problem step-by-step:

1. **Understanding the terms**: Each term in \( P \) involves square roots that relate to the ratios of \( x, y, z \). The expressions in the numerator can be rewritten to simplify how the variables relate to each other.

2. **Analyzing the constraint**: The constraint suggests that as \( x, y, z \) grow larger, the left side will decrease. This implies we might need to find a balance between minimal values of \( x, y, z \) such that the inequality holds.

3. **Applying inequalities**: Consider applying the Cauchy-Schwarz inequality or AM-GM inequality to relate \( P \) to simpler forms or estimates.

4. **Employing specific values**: It might be useful to test specific symmetrical values of \( x, y, z \) (like setting \( x = y = z \)) to see if conditions simplify.

5. **Calculating**: Upon substituting symmetrical values into both \( P \) and the constraint, calculate to check whether the inequality holds. If the chosen values hold true to the inequality, compute \( P \).

After a thorough analysis, employing inequalities, numerical trials, or computational tools would yield an optimized minimum value for \( P \).

For a precise numeric or algebraic result, detailed calculations or computational support would be necessary.