University of Mississippi High School Mathematics Contest – Team Competition March 24, 2006 1. Customers arrive randomly at a rate of x per hour at a ticket counter, and according to a model of the ticket line when one attendant is working, the average length of the line as a x2 where 0 ≤ x < 20. It is decided that it average line function of x is given by f (x) = 400−20x length should not exceed 9 people. For what values of x will a second attendant need to be employed? 2. For three of the four following letter sequences, find the number of ways you can arrange the letters in the sequence so that no two adjacent letters are the same. For example, MPSISISISIP would be one suitable rearrangement of the letters of Mississippi. MISSISSIPPI FLORIDA ALABAMA ALASKA 3. If you collect change in a change drawer over several months, what fraction of the coins would be expected to be dimes? Assume the probability of receiving any value of change between 0 and 99 cents is the same for each purchase, and assume that the clerk always makes change in the most efficient manner possible. 4. How many integer values may a+1 a + a+1 ab take on, where a and b are integers? 5. The rectangles in a set S may be arranged to form a square. There exists a subset of S that consists of three squares, no two of which are congruent to each other. What is the smallest possible number of rectangles in S? Explain why there may be no fewer. 6. A circular pizza is divided using six cuts as shown in the diagram, so that the two perpendicular diameters are each divided into four equal parts by parallel cuts. How many times larger is one of the the largest pieces than one of the smallest ones? Express your answer in terms of π. 7. Four flattened colored cubes are shown below. Each of the cubes’ faces has been colored red (R), blue (B), green (G), or yellow (Y). The cubes are stacked on top of each other in numerical order with cube #1 on bottom. The goal is to find an orientation of each cube so that on each of the four visible sides of the stack all four colors appear. Find a solution, and for each side of the stack, list the colors from bottom to top. List the sides in clockwise order. R G G B R Y G B R R B Y B B R Y G Y R B R Y G Y 8. Note: n!, read n factorial, is defined by n! = n · (n − 1) · (n − 2) · · · · · 2 · 1 (a) How many primes numbers are in the following sequence? 8!+2, 8!+3, 8!+4, 8!+5, 8!+6, 8!+7, 8!+8 (b) Show that no matter how large a positive integer N is, we may find a string of N consecutive composite numbers.