UNIVERSITY OF MARYLAND MATHEMATICS COMPETITIONPART I, 1998No calculators are allowed. 75 min.For each of the following questions, carefully blacken the appropriate box on the answer sheet with a #2 pencil. Do not fold, bend or write stray marks on either side of the answer sheet. Each correct answer is worth 4 points. Two points are deducted for each incorrect answer. Zero points are given if no box, or more than one box, is marked. Note that wild guessing is apt to lower your score.1. Five candidates ran for the office of dog catcher. No two had the same number of votes. The winner had 10 votes, What is the largest possible number of votes for the candidate who finished last? 3, 4, 5, 6, 7

2. How many of the numbers 555555, 5555555, 55555555, 555555555 and 5555555555 are divisible by 9? 0, 1, 2, 3, none of the preceding

3. Rank the following 4 numbers from smallest to largest: A=(-2)^{-2}, B=(-3)^{-3}, C=(-4)^{-4}, D=-77^{1/77 }ABCD, DBCA, BCAD, DABC, none of the preceding

4. An old fashioned toaster can toast one side of up to 4 slices of bread in one minute. What is the least time required to toast both sides of 9 slices? 4, 5, 6, 7, none of the preceding

5. In a restaurant a certain main course costs $22.50 more than the dessert. The main course and the dessert together cost 12 times the price of the dessert. The cost in dollars of dessert is in the range of (1.90, 2.10), (2.10, 2.30), (2.30, 2.50), (2.50, 4.00), more than 4.00

6. The largest numbernso that 8divides 44^{n}^{44}is 8,22,29,44,88

7. A=7/8, B=66/77, C=555/666, D=4444/5555, E=33333/44444. Which is the largest? A, B, C, D, E

8. Five girls A,B,C,D,E sit on 2 chairs and 3 stools, each seating exactly one girl. Who sits on the chairs if A and B sit on the same type of seat, B and D sit on a different type, D and E sit on a different type? CD, AD, BE, BC, AB

9. A (12-hour) wristwatch is slow. It loses 5 minutes per day. Assuming that it is now set to the correct time, how long will it be before it next shows the correct time? 2400 minutes, 60 hours, 80 days, 144 days, 240 days

10. For which value ofadoes the straight liney=6xintersect the parabolay=xin exactly one point? 7, 8, 9, 10, 11^{2}+a

11. To move a heavy safe Donald Duck puts 2 cylinders 5 inches in diameter underneath the safe. The cylinders make one complete revolution. The distance in inches that the safe moves forward is in the interval (15, 30), (30, 45), (45, 60), (60, 75), none of the preceding

12. Leta=log3 andb=log3/(log(log3)) (logarithms are to base 10). Which of the following is equal toa? 1/log3, 1, log3, 1/3, 3^{b}

13. Letf(x)=(x+a). How many pairs of real numbers^{3}+b(a,b)are there such thatf(0)=1andf(1)=2? 0, 1, 2, 3, 4

14. Jack and Jill went up the hill at 4 mph. They started tumbling down at 6 mph. Unfortunately, they hit a rock (and broke their crowns) at exactly halfway down the hill. What was their average speed in mph during the trip up and halfway down? 4.5, 14/3, 4.8, 5, 5.4

15. If ABCDE is a regular pentagon, then the angle ACE, measured in degrees, is in the interval 0-15, 16-30, 31-45, 46-60, 61-90

16. Ifnis the number of integers between 1 and 999 that have at least one 7 in their decimal representation, then 100<n<=150, 150<n<=200, 200<n<=250, 250<n<=300, 300<n<=350

17. Dumbbells weigh 20, 30 or 40 lbs. The total weight of a pile of dumbbells is 800 lbs. The number of dumbells in the pile that weigh 30 lbs can NOT be 2,3,4,6,10

18. The digits 1, 9, 9, 8 in 1998 have their total 1+9+9+8=27. The next time the sum of the digits is 27 happens between the years 2500 and 2700, 2701 and 2900, 2901 and 3100, 3101 and 9900, 9901 and 9999

19. What is the maximal number of pieces into which the regiony=>xcan be cut by 4 straight lines? 8,9,10,11, none of the preceding^{2}

20. The number (7+4·3^{1/2})^{1/2}+ (7-4·3^{1/2})^{1/2}is closest to which of the following: 3.7, 3.8, 3.9, 4.0, 4.1

21. Consider the triangular array 1 2 3 4 5 6 7 8 9 10 . . . . . . . . . . . The sum of the elements in the 100th row is 1000100, 1000000, 500000, 500050, 5000050

22. A train travels at speedsand a runner runs at speed_{1 }s_{2}. It takest_{1}seconds for the entire train to pass the runner when they are going in the same direction andt_{2 }seconds when they are going in opposite directions. Supposes_{1}/s_{2}=t_{1}/t_{2}. Thens_{1}/s_{2}is closest to 3, 2.5, 2, 1.5, 1

23. A quadratic polynomialp(x)satisfiesp(0)=3,p(1)=5,p(2)=8. Thenp(5) is 22, 23, 24, 25, none of the preceding

24. The angle between the hour hand and the minute hand is measured every minute, beginning at 12:01 and ending at 11:59. The smallest angle observed is 5.5^{o}, 3^{o},1^{o}, 0.5^{o}, 0.25^{o }25. The length of one side of a regular tetrahedron inscribed in a sphere of unit radius is closest to 1.1, 1.3, 1.5, 1.7, 1.9