Let $a_1$, $a_2$, $a_3$, $a_4$, and $a_5$ be real numbers satisfying
\begin{align*}
a_1a_2 + a_2a_3 + a_3a_4 + a_4a_5 + a_5a_1 & = 20,\\
a_1a_3 + a_2a_4 + a_3a_5 + a_4a_1 + a_5a_2 & = 22.
\end{align*}Then the smallest possible value of $a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2$ can be expressed as $m + \sqrt{n}$, where $m$ and $n$ are positive integers. Compute $100m + n$.

[i]Proposed by Ankan Bhattacharya[/i]
To find the smallest possible value of \( a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2 \) given the constraints:
\[
a_1a_2 + a_2a_3 + a_3a_4 + a_4a_5 + a_5a_1 = 20,
\]
\[
a_1a_3 + a_2a_4 + a_3a_5 + a_4a_1 + a_5a_2 = 22,
\]
we will use the method of Lagrange multipliers.

1. **Define the functions:**
   \[
   f(a) = a_1 + a_2 + a_3 + a_4 + a_5,
   \]
   \[
   g(a) = a_1a_2 + a_2a_3 + a_3a_4 + a_4a_5 + a_5a_1,
   \]
   \[
   h(a) = a_1a_3 + a_2a_4 + a_3a_5 + a_4a_1 + a_5a_2.
   \]

2. **Express the sum of squares:**
   \[
   \left(\sum_{i=1}^5 a_i\right)^2 = \sum_{i=1}^5 a_i^2 + 2\sum_{1 \le i < j \le 5} a_ia_j.
   \]
   Using the given constraints, we have:
   \[
   \sum_{1 \le i < j \le 5} a_ia_j = g(a) + h(a) = 20 + 22 = 42.
   \]
   Therefore,
   \[
   \left(\sum_{i=1}^5 a_i\right)^2 = \sum_{i=1}^5 a_i^2 + 84 \implies \sum_{i=1}^5 a_i^2 = f(a)^2 - 84.
   \]

3. **Use Lagrange multipliers:**
   If \( f \) has an extremum at \( a \in V \), then \( \nabla f(a) = x \nabla g(a) + y \nabla h(a) \) for some \( x, y \in \mathbb{R} \). This gives us the system:
   \[
   x \begin{pmatrix} a_2 + a_5 \\ a_3 + a_1 \\ a_4 + a_2 \\ a_5 + a_3 \\ a_1 + a_4 \end{pmatrix} + y \begin{pmatrix} a_3 + a_4 \\ a_4 + a_5 \\ a_5 + a_1 \\ a_1 + a_2 \\ a_2 + a_3 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \\ 1 \\ 1 \\ 1 \end{pmatrix}.
   \]

4. **Solve the system:**
   Adding all 5 equations results in:
   \[
   2(x + y)(a_1 + a_2 + a_3 + a_4 + a_5) = 5.
   \]
   Hence, \( x + y \neq 0 \). If there is a unique solution, then \( a_i = \frac{1}{2(x+y)} \), but this implies \( g(a) = h(a) \), which contradicts \( g(a) = 20 \) and \( h(a) = 22 \). Therefore, \( \det M = 0 \).

5. **Find the determinant:**
   \[
   0 = \det M = 2x^5 - 10x^4y + 10x^3y^2 + 10x^2y^3 - 10xy^4 + 2y^5 = 2(x+y)(x^2 - 3xy + y^2)^2.
   \]
   Since \( x + y \neq 0 \), we have:
   \[
   x^2 - 3xy + y^2 = 0 \implies y = \left(\frac{3 \pm \sqrt{5}}{2}\right)x.
   \]

6. **Parameterize and solve:**
   Using the positive root \( y = \left(\frac{3 + \sqrt{5}}{2}\right)x \), we find:
   \[
   \phi = \frac{1 - \sqrt{5}}{2}.
   \]
   We parameterize:
   \[
   a_1 = u + \phi s - t, \quad a_2 = \phi u + s + t, \quad a_3 = u - s + \phi t, \quad a_4 = -\phi s, \quad a_5 = -\phi t.
   \]
   Simplifying, we find:
   \[
   \sum_{i=1}^5 a_i^2 = (2+\phi)^2 \left(\frac{20\phi^2 + 22}{2\phi^3 + 1}\right) - 84 = 21 + \sqrt{5}.
   \]

The final answer is \( \boxed{ 100m + n = 100 \cdot 21 + 5 = 2105 } \).