UNIVERSITY OF MARYLAND MATHEMATICS COMPETITION
PART I, 1998
No 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=-771/77
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 number n so that 8n divides 4444 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 of a does the straight line
y=6x intersect the parabola y=x2 +a in exactly one point?
7, 8, 9, 10, 11
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. Let a=log3 and b=log3/(log(log3)) (logarithms are
to base 10). Which of the following is equal to ab ?
1/log3, 1, log3, 1/3, 3
13. Let f(x)=(x+a)3+b. How many pairs of real numbers
(a,b) are there such that f(0)=1 and f(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. If n is 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 region
y=>x2 can be cut by 4 straight lines?
8,9,10,11, none of the preceding
20. The number (7+4·31/2)1/2 + (7-4·31/2)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 speed s1 and a runner runs at speed s2.
It takes t1 seconds for the entire train to pass the runner when
they are going in the same direction and t2 seconds when they are
going in opposite directions. Suppose s1/s2=t1/t2. Then
s1/s2 is closest to
3, 2.5, 2, 1.5, 1
23. A quadratic polynomial p(x) satisfies p(0)=3, p(1)=5, p(2)=8.
Then p(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
5.5o, 3o,1o, 0.5o, 0.25o
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
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