Senior Exam Page 1 of 5

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1 Senior Exam Page of 5. Triangle ABC has area 80. and AK = KM = MO = OB AL = LN = NP = P C. Find the area of quadrilateral MNP O. B O 20 (2) (4) 30 M 2. Evaluate K A log (2) 00 3 (4) On my bookshelf I have a copy of the four-volume set The World of Mathematics. I decide to take all four volumes down and put them back in such a way that none is in its proper place. In how many different ways can I accomplish this? 4 (2) 5 6 (4) 24 L N P C 4. Among the dogs owners at the Furman University Kennel Club dog show are Mr. Basset, Miss Beagle, Mr. Shepherd and Mr. Spitz, each of whom is the namesake of a breed of dog brought by one of the other three. In an altercation with the shepherd, Mr. Spitz dog bit the shepherd s owner s wife. The basset s owner kept his dog well away from the scuffle. Who owns the spitz? Mr. Shepherd (2) Miss Beagle Mr. Basset (4) Mr. Spitz 5. What is the sum of the squares of the roots of x 4 5x = 0? 9 (2) 0 (4) 2 6. A certain three-digit number exceeds the sum of its digits by 26. The three-digit number obtained by reversing the order of the digits exceeds the sum of the digits by 225. What is the sum of the digits of this number? 6 (2) 7 8 (4) 9 7. You have probably heard that you can determine the (Fahrenheit) temperature by counting the number of chirps a cricket makes in 5 seconds and adding 39. Recently, it has been discovered that the activity of certain ants is also affected by temperature and, in fact, the temperature in Fahrenheit degrees is 39 more than the number of inches the ant travels in minutes. How many times did the cricket chirp while the ant traveled 0 inches across my picnic table? 400 (2) (4) If s men working s hours a day complete a job in s days, how many days would the job take t men working t hours per day? s t t 2 s 3 (2) s3 t 2 (4) s2 t

2 Senior Exam Page 2 of 5 9. When Phileas Fogg completed his epic 80-day circumnavigation, one of his adversaries at the Reform Club registered an objection. After all, Fogg, you did not travel a great circle route, such as the equator. Indeed, you averaged 30 north latitude. Assuming it takes Fogg 80 days to go around the world at 30 north latitude, how long would it take him at the same speed to go around the equator? (2) (4) Find the area of the region consisting of all points (x, y) so that x + y 2. 4 (2) π + 5 (4) 6. Find the sum of all the numbers in all of the first 20 rows of Pascal s triangle.,040,322 (2),048, ,287 (4) 2,097,5 2. What is the unit s digit in ( ) ? 2 (2) 4 6 (4) 8 3. A certain complex number satisfies What is ω 99? ω 2 = ω. (2) ω ω (4) 4. How many integers less than 000 have no factors (other than ) in common with 000? 400 (2) 40 4 (4) What is the slope of the line of positive slope which bisects the angle formed by the lines y = x and y = 7x? 3.5 (2) 3 4 (4) Let N = 999, 999, 999, 999, 999, 999, 999. How many times does the digit 9 appear in the standard representation of N 2? 9 (2) 20 2 (4) For how many integers N is N 4 + 6N < 6N 3 + N 2? (2) 2 3 (4) 4 8. Assuming that two teams are evenly matched (each has probability 2 of winning any game), what is the probability that the World Series will require the full seven games? (The series goes on until one team has won four games.) 2 (2) (4) If A = 2 35, B = 5 5 and C = 6 4, arrange these in increasing order. ABC (2) BCA CAB (4) ACB

3 Senior Exam Page 3 of Let the solution of x + x + x x = 3 x x + x be written as x = a b, where this fraction is in lowest terms. What is a + b? 5 (2) 9 2 (4) Without calculator or tables (which you shouldn t have access to anyway) evaluate cos 87 sin 87 sin cos. (2) 2 3 (4) A conical paper cup is constructed by gluing together the straight sides, AB and AC in the figure below, which is a sector of a circle. Suppose that AB = 3 and A = 20. The volume of water the cup can hold is represented by a π 2 3. What is a? 23. Find the sum (2) (4) Back when I lived in Indiana, my neighbor and I jointly invested in a snow blower. During one January storm, it began to snow sometime during the night and it kept up at a uniform rate throughout the following day. At 6:00 on the morning of the storm, I started the snow blower, which removes snow at a constant rate (in cubic feet per minute), and it took a half hour to clear my driveway (at least momentarily) of snow. Although my neighbor s driveway was only 2 3 as long as mine, he didn t start the blower until 8:00 AM and it took him a half hour also. At what time did the snow start to fall? (You may assume that my neighbor s driveway is the same width as mine.) 3:00 AM (2) 3:30 AM 3:45 AM (4) 4:00 AM B 3 3 C 25. Find the sum of the digits of the largest positive integer which will leave the same remainder when it is divided into 99 or 4 or 204. (2) 2 3 (4) 4 20 A (2) 2 3 (4) Suppose that B is a subset of A. There are precisely 2 subsets of A which are not also subsets of B. How many members does set A have? 4 (2) 5 6 (4) 7

4 Senior Exam Page 4 of In the Fibonacci sequence,, 2, 3, 5, 8, 3, 2, 34, 55,... each term after the second is the sum of the preceding two. The 49th and 50th terms are 7, 778, 742, 049 and 2, 586, 269, 025 respectively. Find the sum of the first 50 terms. 32,95,280,095 (2) 32,95,280,096 32,95,280,097 (4) 32,95,280, A circle is inscribed in a quadrilateral ABCD. AB = 4, BC = 5, CD = 8. Find DA. 7.5 (2) (4) When a clock moves through space at a speed v meters per second for t seconds, then, according to Einstein s special theory of relativity, the clock registers only t v2 c 2 seconds where c, the speed of light, is approximately meters per second. A fast sprinter does the 00 meters in 0 seconds. Approximate to one significant place the difference between the stationary referee s watch and that of the sprinter. More specifically, if this difference is ɛ, find the largest positive integer n, and the smallest positive integer d with d 9, so that ɛ < d 0 n. d = 5, n = 5 (2) d = 6, n = 5 d = 5, n = 6 (4) d = 6, n = If ABCDEF is a regular hexagon, what is the degree measure of AOD, where O is the intersection of the lines segments AE and F D. 5 (2) (4) Find an acute angle A which satisfies 2 cos A = (2) (4) A sphere of radius one is inscribed in a cube. A smaller sphere is inscribed in one of the corners, tangent to the sphere and to three faces of the cube. Find the radius of the smaller sphere. 3 (2) (4) 2 3

5 Senior Exam Page 5 of 5 Bonus Questions: Show all your work. The solution to No. should be written on the green sheet labeled 4, and the solution to No. 2 should be written on the red sheet labeled 42. These should be available from your proctor.. Evaluate Show your work Cylindrical oil drums one meter in diameter are being brought into a 200 meter by 200 meter storage yard and packed together as shown. Approximate the number of barrels that will fit in the yard. 200 m 200 m

8 th Grade Exam Scoring Format: 3 points per correct response -1 each wrong response 0 for blank answers

Pellissippi State Middle School Mathematics Competition 8 th Grade Exam Scoring Format: 3 points per correct response -1 each wrong response 0 for blank answers Directions: For each multiple-choice problem