Numbers $m$ and $n$ are given positive integers. There are $mn$ people in a party, standing in the shape of an $m\times n$ grid. Some of these people are police officers and the rest are the guests. Some of the guests may be criminals. The goal is to determine whether there is a criminal between the guests or not.\\
    Two people are considered \textit{adjacent} if they have a common side. Any police officer can see their adjacent people and for every one of them, know that they're criminal or not. On the other hand, any criminal will threaten exactly one of their adjacent people (which is likely an officer!) to murder. A threatened officer will be too scared, that they deny the existence of any criminal between their adjacent people.\\
    Find the least possible number of officers such that they can take position in the party, in a way that the goal is achievable. (Note that the number of criminals is unknown and it is possible to have zero criminals.)

[i]Proposed by Abolfazl Asadi[/i]
To solve this problem, we need to determine the minimum number of police officers required to ensure that any criminal present in the party can be detected. The key idea is to use a checkerboard pattern to place the police officers. Let's break down the solution step-by-step:

1. **Checkerboard Coloring**:
   - Consider the \(m \times n\) grid. We can color the grid in a checkerboard pattern such that no two adjacent cells have the same color. This means that each cell is either white or black, and each white cell is adjacent only to black cells and vice versa.

2. **Placing Police Officers**:
   - Place police officers on all cells of one color (say white). This ensures that every black cell (which contains guests) is adjacent to at least one white cell (which contains a police officer).

3. **Counting the Number of Police Officers**:
   - In a checkerboard pattern, approximately half of the cells will be white and half will be black. If \(mn\) is even, exactly half of the cells will be white. If \(mn\) is odd, one more than half of the cells will be white.
   - Therefore, the number of police officers required is \(\left\lceil \frac{mn}{2} \right\rceil\).

4. **Ensuring Detection of Criminals**:
   - Any criminal in a black cell will threaten one of its adjacent white cells. Since each white cell is a police officer, the police officer will be able to detect the criminal unless they are threatened.
   - If a police officer is threatened, they will deny the existence of a criminal. However, since we have placed police officers on all white cells, there will always be at least one police officer adjacent to any criminal who is not threatened. This ensures that the criminal can be detected.

5. **Verification**:
   - To verify that this is the minimum number of police officers required, consider the case where there are fewer than \(\left\lceil \frac{mn}{2} \right\rceil\) police officers. In this case, there will be at least one black cell (guest) that is not adjacent to any white cell (police officer). This means that a criminal in this black cell cannot be detected, which contradicts the goal.

Therefore, the least possible number of police officers required to ensure that any criminal can be detected is \(\left\lceil \frac{mn}{2} \right\rceil\).

The final answer is \( \boxed{ \left\lceil \frac{mn}{2} \right\rceil } \).