Download university of maryland mathematics competition

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
22.
23.
24.
25.
1000100, 1000000, 500000, 500050, 5000050
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
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
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
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
UNIVERSITY OF MARYLAND MATHEMATICS COMPETITION
PART II, 1998
No calculators are allowed.
PROOFS MUST BE GIVEN FOR ALL ANSWERS
1. Four positive numbers are placed at the vertices of a rectangle. Each number is at least
as large as the average of the two numbers at the adjacent vertices. Prove that all four
numbers are equal.
2. The sum 498 + 499 + 500 + 501 = 1998 is one way of expressing 1998 as a sum of
consecutive positive integers. Find all ways of expressing 1998 as a sum of two or more
consecutive positive integers. Prove your list is complete.
3. An infinite strip (two parallel lines and the region between them) has a width of 1 inch.
What is the largest value of A such that every triangle with area A square inches can be
placed on this strip? Justify your answer.
4. A plane divides space into two regions. Two planes that intersect in a line divide space
into four regions. Now suppose that twelve planes are given in space so that a) every two
of them intersect in a line, b) every three of them intersect in a point, and c) no four of
them have a common point. Into how many regions is space divided? Justify your answer.
5. Five robbers have stolen 1998 identical gold coins. They agree to the following: The
youngest robber proposes a division of the loot. All robbers, including the proposer, vote
on the proposal. If at least half the robbers vote yes, then that proposal is accepted. If not,
the proposer is sent away with no loot and the next youngest robber makes a new
proposal to be voted on by the four remaining robbers, with the same rules as above. This
continues until a proposed division is accepted by at least half the remaining robbers.
Each robber guards his best interests: He will vote for a proposal if and only if it will give
him more coins than he will acquire by rejecting it, and the proposer will keep as many
coins for himself as he can. How will the coins be distributed? Explain your reasoning.