If the sum of the coefficients in the expansion of
`(b + c)^(20) {1 +(a -2) x}^(20)` is equal to square of the sum of the
coefficients in the expansion of `[2 bcx - (b + c)y]^(10)`, where a, b, c are
positive constants, then

Options

A. ` ge sqrt((a c)`
B. `(b +c)/(2) ge a`
C. c, a and b are in G. P
D. `(1)/(c),(1)/(a),(1)/(b)` are in H.P

Explanation

To solve the problem step by step, we will analyze the given expressions and find the required relationships.

### Step 1: Find the sum of coefficients in the expansion of \((b + c)^{20} (1 + (a - 2)x)^{20}\)

To find the sum of the coefficients in the expansion of any polynomial, we substitute \(x = 1\).

1. Substitute \(x = 1\) in \((1 + (a - 2)x)^{20}\):
\[ (1 + (a - 2) \cdot 1)^{20} = (1 + a - 2)^{20} = (a - 1)^{20} \]

2. Now, substitute \(x = 1\) in \((b + c)^{20}\):
\[ (b + c)^{20} \]

3. Therefore, the sum of coefficients \(S_1\) is:
\[ S_1 = (b + c)^{20} \cdot (a - 1)^{20} \]

### Step 2: Find the sum of coefficients in the expansion of \([2bcx - (b + c)y]^{10}\)

Again, to find the sum of coefficients, we substitute \(x = 1\) and \(y = 1\).

1. Substitute \(x = 1\) and \(y = 1\):
\[ [2bc \cdot 1 - (b + c) \cdot 1]^{10} = [2bc - (b + c)]^{10} = (2bc - b - c)^{10} \]

2. Therefore, the sum of coefficients \(S_2\) is:
\[ S_2 = (2bc - b - c)^{10} \]

### Step 3: Set up the equation based on the problem statement

According to the problem, the sum of the coefficients \(S_1\) is equal to the square of the sum of coefficients \(S_2\):
\[ (b + c)^{20} \cdot (a - 1)^{20} = \left((2bc - b - c)^{10}\right)^2 \]
This simplifies to:
\[ (b + c)^{20} \cdot (a - 1)^{20} = (2bc - b - c)^{20} \]

### Step 4: Simplify the equation

Since both sides are raised to the power of 20, we can take the 20th root:
\[ (b + c)(a - 1) = 2bc - (b + c) \]

### Step 5: Rearrange the equation

Rearranging the equation gives:
\[ (b + c)(a - 1) + (b + c) = 2bc \]
\[ (b + c)(a) = 2bc \]

### Step 6: Solve for \(a\)

From the equation:
\[ a = \frac{2bc}{b + c} \]

### Step 7: Identify the relationship

This shows that \(a\) is the harmonic mean of \(b\) and \(c\). By the properties of means, we know that the arithmetic mean is always greater than or equal to the harmonic mean:
\[ \frac{b + c}{2} \geq a \]

### Conclusion

Thus, we conclude that \(a\) is the harmonic mean of \(b\) and \(c\), and the final relationship is:
\[ \frac{b + c}{2} \geq a \]