Each of the numbers `x_1,x_2,x_3,x_4,x_5,...,x_n` `x_nge4` is equal to -1 or 1. If
`x_1x_2x_3+x_2x_3x_4+x_3x_4x_5+...x_(n-3)x_(n-2)x_(n-1)+x_(n-2)x_(n-1)x_n=0` the minumum number of `x_i`(i.e., `x_1,x_2,x_3,…x_n`) are equal to -1 is:

Options

A. `n/3`
B. `n/2`
C. `(n-2)/2`
D. can't be determined

Answer

To solve the problem, we need to analyze the expression given and determine the minimum number of \( x_i \) that must be equal to -1 in order for the entire expression to equal 0.

Given:
- Each \( x_i \) can be either -1 or 1.
- The expression is:
\[ x_1 x_2 x_3 + x_2 x_3 x_4 + x_3 x_4 x_5 + \ldots + x_{n-2} x_{n-1} x_n = 0 \]

### Step 1: Understanding the Expression
The expression consists of a sum of products of three consecutive \( x_i \) values. Each product \( x_i x_{i+1} x_{i+2} \) can either be -1 or 1 depending on the values of \( x_i \).

### Step 2: Analyzing the Products
- If all three \( x_i \) are 1, then the product is 1.
- If one of the \( x_i \) is -1 and the other two are 1, the product is -1.
- If two of the \( x_i \) are -1 and one is 1, the product is 1.
- If all three \( x_i \) are -1, the product is -1.

### Step 3: Setting Up the Equation
For the entire sum to equal 0, the number of products that yield 1 must equal the number of products that yield -1.

### Step 4: Counting the Products
There are \( n - 2 \) terms in the sum (from \( x_1 x_2 x_3 \) to \( x_{n-2} x_{n-1} x_n \)).

Let:
- \( p \) = number of terms that equal 1
- \( q \) = number of terms that equal -1

From the equation:
\[ p + q = n - 2 \]
And since we need \( p = q \) for the sum to be zero:
\[ p = q \]
Thus:
\[ p + p = n - 2 \implies 2p = n - 2 \implies p = \frac{n - 2}{2} \]
This means:
\[ q = \frac{n - 2}{2} \]

### Step 5: Relating to the Values of \( x_i \)
Each term that contributes -1 requires at least one \( x_i \) to be -1. Therefore, to achieve \( q \) terms of -1, we need at least \( q \) values of \( x_i \) to be -1.

### Step 6: Finding the Minimum Number of -1's
Since \( q = \frac{n - 2}{2} \), the minimum number of \( x_i \) that must be -1 is:
\[ \text{Minimum number of } x_i = \frac{n - 2}{2} \]

### Step 7: Considering the Remaining Terms
However, since we also need to ensure that the total number of \( x_i \) is at least 4 (as \( n \geq 4 \)), we can conclude that:
- For \( n = 4 \), the minimum is 1.
- For \( n = 5 \), the minimum is 1.5 (round up to 2).
- For larger \( n \), the minimum number of -1's will always be \( \frac{n}{2} \).

### Final Conclusion
Thus, the minimum number of \( x_i \) that must be -1 is:
\[ \boxed{\frac{n}{2}} \]