Let $\ell$ be a line and let points $A$, $B$, $C$ lie on $\ell$ so that $AB = 7$ and $BC = 5$. Let $m$ be the line through $A$ perpendicular to $\ell$. Let $P$ lie on $m$. Compute the smallest possible value of $PB + PC$. 

[i]Proposed by Ankan Bhattacharya and Brandon Wang[/i]
1. **Identify the given information and setup the problem:**
   - Line $\ell$ contains points $A$, $B$, and $C$ such that $AB = 7$ and $BC = 5$.
   - Line $m$ is perpendicular to $\ell$ and passes through point $A$.
   - Point $P$ lies on line $m$.
   - We need to find the smallest possible value of $PB + PC$.

2. **Visualize the problem:**
   - Draw line $\ell$ with points $A$, $B$, and $C$ in a straight line.
   - Draw line $m$ perpendicular to $\ell$ at point $A$.
   - Point $P$ can be any point on line $m$.

3. **Use geometric properties:**
   - Since $m$ is perpendicular to $\ell$ at $A$, any point $P$ on $m$ will form right angles with $\ell$.
   - The distance from $P$ to any point on $\ell$ can be considered as the hypotenuse of a right triangle where one leg is the vertical distance from $P$ to $\ell$ and the other leg is the horizontal distance along $\ell$.

4. **Calculate distances:**
   - Let $P$ be at a distance $d$ from $A$ on line $m$.
   - The distance $PB$ can be calculated using the Pythagorean theorem: 
     \[
     PB = \sqrt{d^2 + AB^2} = \sqrt{d^2 + 7^2} = \sqrt{d^2 + 49}
     \]
   - Similarly, the distance $PC$ is:
     \[
     PC = \sqrt{d^2 + AC^2}
     \]
     where $AC = AB + BC = 7 + 5 = 12$, so:
     \[
     PC = \sqrt{d^2 + 12^2} = \sqrt{d^2 + 144}
     \]

5. **Minimize the sum $PB + PC$:**
   - We need to find the minimum value of:
     \[
     PB + PC = \sqrt{d^2 + 49} + \sqrt{d^2 + 144}
     \]
   - To minimize this expression, consider the case when $d = 0$ (i.e., $P$ is at point $A$):
     \[
     PB + PC = \sqrt{0 + 49} + \sqrt{0 + 144} = \sqrt{49} + \sqrt{144} = 7 + 12 = 19
     \]

6. **Check if there is a smaller value:**
   - Since $P$ is on line $m$, moving $P$ away from $A$ will increase both $PB$ and $PC$ due to the Pythagorean theorem.
   - Therefore, the minimum value occurs when $P$ is at $A$.

The final answer is $\boxed{19}$.