Let $n$ be a positive integer and $a_1, a_2, \dots, a_n$ non-zero real numbers. What is the least number of non-zero coefficients that the polynomial $P(x) = (x - a_1)(x - a_2)\cdots(x - a_n)$ can have?
1. **Understanding the Problem:**
   We need to determine the least number of non-zero coefficients that the polynomial \( P(x) = (x - a_1)(x - a_2) \cdots (x - a_n) \) can have, where \( a_1, a_2, \ldots, a_n \) are non-zero real numbers.

2. **Analyzing the Polynomial:**
   The polynomial \( P(x) \) is a product of \( n \) linear factors. When expanded, it will be a polynomial of degree \( n \) with the general form:
   \[
   P(x) = x^n + c_{n-1}x^{n-1} + c_{n-2}x^{n-2} + \cdots + c_1x + c_0
   \]
   where \( c_i \) are the coefficients.

3. **Counting Non-Zero Coefficients:**
   We need to find the minimum number of non-zero coefficients \( c_i \). 

4. **Using Symmetry and Patterns:**
   Consider the polynomial \( P(x) = (x - a_1)(x - a_2) \cdots (x - a_n) \). The coefficients \( c_i \) are symmetric sums of the roots \( a_1, a_2, \ldots, a_n \). Specifically, each \( c_i \) is a sum of products of \( n-i \) roots taken at a time.

5. **Using Combinatorial Argument:**
   The number of non-zero coefficients can be minimized by choosing the roots \( a_i \) in such a way that many of these symmetric sums become zero. 

6. **Applying Known Results:**
   It is known that there exist at least \( (n+1) - \lceil (n-1)/2 \rceil \) non-zero coefficients. This result can be derived from combinatorial arguments and properties of symmetric sums.

7. **Case Analysis:**
   - For even \( n \), the equality case holds when, for example, \( a_k = (-1)^k \lceil k/2 \rceil \) for \( k = 1, 2, \ldots, n \).
   - For odd \( n \), finding an explicit example is more challenging, but the same lower bound applies.

8. **Conclusion:**
   The least number of non-zero coefficients that the polynomial \( P(x) \) can have is given by:
   \[
   (n+1) - \left\lceil \frac{n-1}{2} \right\rceil
   \]

The final answer is \( \boxed{ (n+1) - \left\lceil \frac{n-1}{2} \right\rceil } \).