Let $M$ be the maximum possible value of $x_1x_2+x_2x_3+\cdots +x_5x_1$ where $x_1, x_2, \cdots x_5$ is a permutation of $(1,2,3,4,5)$ and let $N$ be the number of permutations for which this maximum is attained. Evaluate $M+N$.
1. **Identify the problem and use the Rearrangement Inequality**:
   We need to find the maximum possible value of \( x_1x_2 + x_2x_3 + x_3x_4 + x_4x_5 + x_5x_1 \) where \( x_1, x_2, x_3, x_4, x_5 \) is a permutation of \( (1, 2, 3, 4, 5) \). The Rearrangement Inequality states that the sum of products of two sequences is maximized when both sequences are similarly sorted.

2. **Arrange the numbers to maximize the sum**:
   To maximize \( x_1x_2 + x_2x_3 + x_3x_4 + x_4x_5 + x_5x_1 \), we should place the largest numbers adjacent to each other. Let's consider the permutation \( (5, 4, 3, 2, 1) \).

3. **Calculate the sum for the permutation \( (5, 4, 3, 2, 1) \)**:
   \[
   5 \cdot 4 + 4 \cdot 3 + 3 \cdot 2 + 2 \cdot 1 + 1 \cdot 5 = 20 + 12 + 6 + 2 + 5 = 45
   \]

4. **Verify if this is the maximum**:
   We need to check if any other permutation can give a higher sum. By the Rearrangement Inequality, the sum is maximized when the largest numbers are adjacent. Any other permutation will result in a lower sum.

5. **Count the number of permutations that achieve this maximum**:
   The maximum sum can be achieved by rotating the permutation \( (5, 4, 3, 2, 1) \) and its reverse \( (1, 2, 3, 4, 5) \). Each of these can be rotated in 5 ways:
   \[
   (5, 4, 3, 2, 1), (4, 3, 2, 1, 5), (3, 2, 1, 5, 4), (2, 1, 5, 4, 3), (1, 5, 4, 3, 2)
   \]
   and
   \[
   (1, 2, 3, 4, 5), (2, 3, 4, 5, 1), (3, 4, 5, 1, 2), (4, 5, 1, 2, 3), (5, 1, 2, 3, 4)
   \]
   Thus, there are \( 5 + 5 = 10 \) permutations that achieve the maximum sum.

6. **Calculate \( M + N \)**:
   \[
   M = 45, \quad N = 10 \quad \Rightarrow \quad M + N = 45 + 10 = 55
   \]

The final answer is \(\boxed{55}\)