REVIEW 1. In the figure shown, a perpendicular segment is drawn from B in rectangle ABCD to meet diagonal AC at point X. Side AB is 6 cm and diagonal AC is 10 cm. How far is point X from the midpoint M of the diagonal AC? Express your answer as a decimal to the nearest tenth. 2. At a graduation ceremony, names were read at the rate of five names per minute. If the names had been read at the rate of seven names per minute, the ceremony would have taken 24 minutes less. How many names were read? 3. On an old-fashioned bicycle the front wheel has a radius of 2.5 feet and the back wheel has a radius of 4 inches. If there is no slippage, how many revolutions will the back wheel make while the front wheel makes 100 revolutions? 4. How many integers from 100 through 999, inclusive, do not contain any of the digits 2, 3, 4 or 5? 5. The first term of a sequence of positive integers is any two-digit integer. Each subsequent term is the sum of the tens digit and the square of the ones digit of the previous term. One possible sequence is 14, 17, 50, 5, 25, … If the first term of one such sequence is 97, what is the 2008th term of the sequence? 6. Two different integers are randomly chosen from the set {-5, -8, 7, 4, -2}. What is the probability that their product is negative? Express your answer as a common fraction. 7. What is the smallest positive integer that is a perfect square and is also a sum of six consecutive positive integers? 8. In the figure shown, AC = 13 and DC – 2 units. What is the length of segment BD? Express your answer in simplest radical form. 9. What is the units digit of the product of 335 x 735? 10. Two bags of marbles are pictured below. One marble is randomly selected from Bag A and placed into Bad B. One marble is then randomly selected from Bag B. What is the probability that the marble selected from Bag B is black? Express your answer as a common fraction. 11. Let P = (2 – 3 – 4 + 7)2347 and Q = (-2 + 3 + 4 – 7) 2347. What is the value of (2 + 3 + 4 + 7)(P + Q)? 12. Suppose that each distinct letter in the equation MATH = COU + NTS is replaced by a different digit chosen from 1 through 9 in such a way that the resulting equation is true. If H = 4, what is the value of the greater of C and N? 13. If = 3.1415926…, what is the exact value of | - 3.14| + | - 22/7|? 14. Two shaded identical rectangular decorative tiles are first placed (one each) at the top and at the base of a door frame for a hobbit’s house, as shown in Figure 1. The distance from W to H is 45 inches. Then the same two tiles are rearranged at the top and at the base of the door frame, as shown in Figure 2. The distance from Y to Z is 37 inches. What is the height of the door frame? 15. Two consecutive even numbers are each squared. The difference of the squares is 60. What is the sum of the original two numbers? 16. The length of a diagonal of a square is sqrt(2) + sqrt(3) unites. What is the area of the square? Express your answer in simplest form as a/b + sqrt(c), where a/b is a common fraction and c has no perfect square factors other than 1. 17. The summary of a survey of 100 students listed the following totals: 59 students did math homework 49 students did English homework 42 students did science homework 20 students did English and science homework 29 students did science and math homework 31 students did math and English homework 12 students did math, science, and English homework How many students did no math, no English and no science homework? 18. There are eight circles in the picture below. The shaded circles are congruent. If each circle in the picture is tangent to each of its neighbor circles, what is the ratio of the total area of the seven shaded regions to the area of the largest circle? Express your answer as a common fraction. 19. How many square units are in the area of the convex quadrilateral with vertices (0, 0), (3, 0), (2, 2) and (0, 3)? 20. Camy made a list of every possible distinct five-digit positive even integer that can be formed using each of the digits 1, 3, 4, 5 and 9 exactly once in each integer. What is the sum of the integers on Camy’s list? 21. How many integers between 1 and 200 are multiples of both 3 and 5 but not of either 4 or 7? 22. Two sides of a square are divided into fourths and another side of the square is trisected, as shown. A triangle is formed by connecting three of these points, as shown. What is the ratio of the area of the shaded triangle to the area of the square? Express your answer as a fraction. 23. When five standard six-sided dice are rolled sequentially there are 65 = 7776 possible outcomes. For how many outcomes is the sum of the five rolled numbers exactly 27? 24. Matt will arrange four identical, dotless, dominoes (shaded 1 by 2 rectangles) on the 5 by 4 grid to the right so that a path is formed from the upper left-hand corner A to the lower right-hand corner B. In a path, consecutive dominoes must touch at their sides and not just their corners. No domino may be placed diagonally; each domino covers exactly two of the unit squares shown on the grid. One arrangement is show. How many distinct arrangements are possible, including the one shown? 25. Regions A, B, C, J and K represent ponds. Logs leave pond A and float down flumes (represented by arrows) to eventually end up in pond B or pond C. On leaving a pond, the logs are equally likely to use any available exit flume. Logs can only float in the direction the arrow is pointing. What is the probability that a log in pond A will end up in pond B? Express your answer as a common fraction. 26. Two Russian mathematicians meet on a plane. “If I remember correctly, you have three sons,” says Ivan. “What are their ages today?” “The product of their ages is thirty-six,” says Igor, “and the sum of their ages is exactly today’s date.” “I’m sorry, Igor,” Ivan says after a minute, “but that doesn’t tell me the ages of your boys.” “Oh, I forgot to tell you, my youngest son has red hair.” “Ah now it’s clear,” Ivan says. “I now know exactly how old your three sons are.” How did Ivan figure out the ages?