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We can solve this problem by considering the number of 1s and 3s used to form the sum 11.

Number of 1s and 3s:

The sum of the digits is 11, which can be formed using digits 1 and 3. Let's say there are x ones and (11 - x) threes.

Restrictions on x:

Since each digit must be a 1 or a 3, x (number of ones) can range from 0 (all digits are threes) to 11 (all digits are ones). However, we only care about positive integers that satisfy the condition. Therefore, valid values for x are:

1 <= x <= 11

Counting the Possibilities:

For each value of x (number of ones), we have (11 - x) threes. So, the total number of positive integers with x ones and (11 - x) threes is 1.

This creates 11 possibilities (from x = 1 to x = 11) since there are 11 valid values for x.

Therefore, there are 11 positive integers that satisfy both conditions.

This functional equation can be solved to find f(x) as a rational function. Here's how:

Isolating f(x): We want to isolate f(x) on one side of the equation. Subtract f((2x - 3)/(x - 1)) from both sides:

f(x) = x - f((2x - 3)/(x - 1))

Finding a Common Denominator: The next step is to find a common denominator for the right side of the equation. Since the function is undefined at x = 1 and x = 2, the common denominator will involve these terms (x - 1) and (x - 2).

We can rewrite f((2x - 3)/(x - 1)) as:

f((2x - 3)/(x - 1)) * (x - 2) / (x - 2) = f((2x - 3)/(x - 1)) * (x - 2) / (x - 1)(x - 2)

This introduces the term (x - 2) in the denominator to match the existing term (x - 1).

Substituting and Combining Like Terms:

Now, substitute the entire right side of the equation back into the expression for f(x):

f(x) = x - [ f((2x - 3)/(x - 1)) * (x - 2) / (x - 1)(x - 2) ]

Expand the product in the denominator:

f(x) = x - [ f((2x - 3)/(x - 1)) * (x - 2) ] / (x^2 - 3x + 2)

Recognizing a Pattern:

Notice that the term f((2x - 3)/(x - 1)) appears in the equation. We can substitute the entire right side of the original equation for this term:

f(x) = x - { [ x - f((2x - 3)/(x - 1)) ] * (x - 2) } / (x^2 - 3x + 2)

This creates a recursive relationship involving f(x). However, it allows us to eliminate f((2x - 3)/(x - 1)) from the equation entirely.

Simplifying and Solving:

Expand the product in the brackets and combine like terms:

f(x) = x - (x^2 - 5x + 2) / (x^2 - 3x + 2)

f(x) = (x^2 - 3x + 2 - x^2 + 5x - 2) / (x^2 - 3x + 2)

f(x) = 2x / (x^2 - 3x + 2)

Therefore, the function f(x) satisfying the given conditions is:

f(x) = 2x / (x^2 - 3x + 2)

There are 487 ways of choosing the three circles.

The answer is E075A.

The answer is 16/93.

N = 407.

Problem 2

Note that both factors share some common terms. Let's rewrite them to group these terms together:

(x^50 + x^48 + ... + x^2 + 1) + (x^49 - x^47 + ... + x - 1)

According to the sum of a finite geometric series formula, the sum of each individual sequence is:

(x^50 + x^48 + ... + x^2 + 1) = (x^(50)) * (1 / (1 - x^2)) = x^(50) / (1 - x^2)

(x^49 - x^47 + ... + x - 1) = (x^(49)) * (1 / (1 + x^2)) = x^(49) / (1 + x^2)

Now we can find the product:

Product = (x^(50) / (1 - x^2)) * (x^(49) / (1 + x^2))

Since we're only interested in the sum of the coefficients, we can ignore the constant terms in the denominators (they won't affect the sum). This simplifies the product to:

Product ≈ x^(50 + 49) // Ignoring constant terms in denominators

The sum of the coefficients in this product is simply the coefficient of the term with x raised to the highest power, which is x^(99). The coefficient of this term is 1.

In conclusion, the sum of the coefficients of the product is 1.

The answer is 17/210.

The answer is 40568/7.

The answer is 51485016_16.