2015 AMC 12A Problems/Problem 24

Problem

Rational numbers and are chosen at random among all rational numbers in the interval that can be written as fractions where and are integers with . What is the probability that is a real number?

$\textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}$ (Error compiling LaTeX. Unknown error_msg)

Solution

First we consider consider the binomial expansion of the expression: (let cos(aπ) be x and sin(bπ) be y) x^4 + i * 4x^3*y + 6x^2*y^2 - i * 4xy^3 + y^4.

We notice that the only terms with i are the second and the fourth terms. Thus for the expression to be a real number, either cos(aπ) or sin(bπ) must be 0, or the second term and the fourth term cancel each other out (because in the fourth term, you have i^2= -1). Let's calculate the probability of the first case: For either cos(aπ) or sin(bπ) to be 0, the two valid a's that would work are 1/2 and 3/2, and the two valid b's that would work are 0 and 1. Because a and b are can both be expressed as fractions with a denominator less or equal to 5, their are a total of 20 possible values for a and b:

0, 1, 1/2, 3/2, 1/3, 2/3, 4/3, 5/3, 1/4, 3/4, 5/4, 7/4, 1/5, 2/5, 3/5, 4/5, 6/5, 7/5, 8/5, 9/5.

We calculate the total number of sets of (a,b) we can find: 20*20=400. We calculate the total number of invalid sets(as in when a doesn't equal 1/2 or 3/2 and b doesn't equal 0 or 1): (20-2)*(20-2) = 324. Thus our number of valid sets is 76.

Let's calculate the probability for the second case: For the two terms to cancel, then

cos(aπ)^3*sin(bπ) = cos(aπ)*sin(bπ)^3

So

cos(aπ)^2 = sin(bπ)^2

Which means for a given value of cos(aπ) or sin(bπ), there are 4 valid values for the corresponding (one in each quadrant). When either cos(aπ) or sin(bπ) is equal to 1 however, there are only two corresponding values. We don't count the sets where either cos(aπ) or sin(bπ) equals 0, for we would get repeated sets. We also exclude values where the denominator is an odd number, for we cannot find any corresponding values. (for example, if a is 1/5, then b must be 3/10, which we don't have) Thus our total number of sets for this case is 4*4 + 2*2 = 20.

Thus our final answer is (20 + 76)/400 = 6/25, which is (D).