How many ordered pairs of positive integers satisfy the equation Solution Problem 11 A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday.

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How many ordered pairs of positive integers satisfy the equation Solution Problem 11 A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible?

Solution Problem 12 Point B is due east of point A. Point C is due north of point B. The distance between points A and C is , and. Point D is 20 meters due north of point C.

The distance AD is between which two integers? Solution Problem 13 It takes Clea 60 seconds to walk down an escalator when it is not operating, and only 24 seconds to walk down the escalator when it is operating. How many seconds does it take Clea to ride down the operating escalator when she just stands on it?

Problem 14 Two equilateral triangles are contained in a square whose side length is. The bases of these triangles are the opposite sides of the square, and their intersection is a rhombus. What is the area of the rhombus? Solution Problem 15 In a round-robin tournament with 6 teams, each team plays one game against each other team, and each game results in one team winning and one team losing.

At the end of the tournament, the teams are ranked by the number of games won. What is the maximum number of teams that could be tied for the most wins at the end on the tournament?

Solution Problem 16 Three circles with radius 2 are mutually tangent. What is the total area of the circles and the region bounded by them, as shown in the figure? Solution Problem 17 Jesse cuts a circular paper disk of radius 12 along two radii to form two sectors, the smaller having a central angle of degrees. He makes two circular cones, using each sector to form the lateral surface of a cone.

What is the ratio of the volume of the smaller cone to that of the larger? Solution Problem 18 Suppose that one of every people in a certain population has a particular disease, which displays no symptoms. A blood test is available for screening for this disease. For a person who has this disease, the test always turns out positive. For a person who does not have the disease, however, there is a false positive rate--in other words, for such people, of the time the test will turn out negative, but of the time the test will turn out positive and will incorrectly indicate that the person has the disease.

Let be the probability that a person who is chosen at random from this population and gets a positive test result actually has the disease. Which of the following is closest to?

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Students must arrive 15 minutes prior to the exam time. AMC 12 — can only be taken by students who are in 12th grade or below and What is probability that the dart lands within the center square? All additional triangles will differ by one as the solutions above differ by one so this process can be repeated indefinately until the side lengths no longer form a triangle. The beauty of mathematics only shows itself to more patient followers. Mark 1, 2 24 Solving equations involving modulo operator I eventually gave up and read the solution posted on the website, but it made sense only up to the halfway mark, when the writer employs an argument about treating a number as a units digit instead of a tens digit.

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