Question: from Matthew Prince, USA
In a certain math contest, there are three categories of questions: Easy worth 3 points each, Medium worth 5 points each, and Hard worth 9 points each. All question types are equally likely to appear. The test consists of 50 questions. The expected total number of points can be written as a reduced mixed number of the form a+b/c. Find a + b + c.
Answer: Since the probabilities of each question being on the test are equal, we can expect 50/3 easy questions, 50/3 medium questions and 50/3 hard questions. Thus we have a total of 3*50/3 + 5*50/3 + 9*50/3= 17*50/3 = 283+1/3. Then, 283 + 1 + 3 = 287
Solution to bonus question
Since x + y = 79, we have ab + b = 9 and 20a + 10b = 70. From the first equation, b = 9/(a + 1). Substituting this into the second equation, we end up with 20a2 - 50b + 20 = 0. Since a is a natural number, a = 2 and b = 3. Then x = 23, y = 56 and xy = 1288.