The First Annual Catonsville Mathematics Competition 1992
- If all the natural numbers are written next to each other as follows
what is the 1992nd digit in the sequence?
A) 0 B) 2 C) 4 D) 7
- If A is is the sum of the digits of 19921992 and B is the sum of the digits of A,
what is the sum of the digits of B?
A) 3 B) 6 C) 9 D) 11
- If you have only lots of 5¢ and 29¢ stamps, what is the largest postage amount you cannot make using these stamps?
A) $0.83 B) $1.11 C) $1.45 D) $2.73
- You are given eight small cubes - two each in red, blue, green, and yellow, all of the same size - and asked to
assemble them into one large cube with each face showing all four colors.
How many different choices of large cubes are possible?
(Note : Two large cubes are considered identical if one of them can be turned so that the positions of the colors
on it matches those on the other.)
A) 1 B) 2 C) 4 D) 8
- Consider three points on a plane that are not on the same line. How many lines can be drawn on the plane that
are each equidistant from the three points?
A) 1 B) 2 C) 3 D) 6
- If a2 + b2 = c2 where a, b, c are positive integers and if
a = 103, then b is
A) 207 B) 672 C) 5304 D) 6702
- A palindromic number is a positive integer which is the same when read from right to left as when read from
left to right. 13731, e.g. is a palindromic number. If M is the number of all 5-digit palindromic numbers
and N is the number of all 6-digit palindromic numbers, what is N/M ?
A) 1 B) 9 C) 10 D) 90
- A file of soldiers one mile long is marching at a constant speed along a straight road on a side of which
stands a pillar. When the front of the file reaches the pillar, the general at the front sends a messenger
to the general at the end of the file with a message. The messenger travels at a constant speed, gives the
message, and promptly turns back, travels at the same speed to the first general and reaches him at the
precise moment when the tail end of the file reaches the pillar.
What distance did the messenger travel in miles?
A) 1 B) Ö2 C) 2 D) Ö2 + 1
- If x = sin(p/7) and y = cot2(p/7)
A) y = (1 - x2)/x2 , B) y = (x2 + 1)/(1 - x2) ,
C) y = 1/(1 - x2) , D) y = x2/(1 - x2)
- The sum of squares of all primes p such that p + 1 is a square is
A) 9 B) 97 C) 373 D) 6250
- A school has 600 students numbered 1 to 600 and a locker for each student also numbered 1 to 600.
On the last day of school all lockers are locked by student no. 1. Then student no. 2 unlocks all
even numbered lockers, after which student no. 3 "unlocks or locks" (i.e., unlocks if locked and
locks if unlocked) all lockers with number divisible by 3 and so on. In the k-th step of this
process, student no. k "unlocks or locks" all lockers with number divisible by k.
Lastly student no. 600 "unlocks or locks" only locker no. 600.
How many lockers remain locked after all this?
A) 24 B) 60 C) 240 D) 300
- A triangle has sides of lengths a, b, c and medians of lengths l, m, n. The ratio
(l2 + m2 + n2)/(a2 + b2 + c2)
A) 1/2 B) 2/3 C) 3/4 D) depends on triangle.
- A triangle has sides of lengths 5, 7, and 8. Its area is
A) 10 B) 10Ö2 C) 10Ö3 D) 12
- A businessman has a wife in Baltimore and a mistress in Washington, D.C. and has his office in Laurel
on the Baltimore-Washington MARC train line, 18 miles from Washington and 27 miles from Baltimore. Trains
run in each direction at 20 minute intervals. The first train every morning leaves Washington at 6 AM,
while the first train from Baltimore leaves at 6:05 AM. The entire trip each way takes 40 minutes.
Everyday the businessman leaves his office in the afternoon and reaches the station at no fixed time but
sometime between 4 PM and 6 PM. He takes the first train that comes in either direction and thus spends
the night with his wife or with his mistress accordingly.
What percent of nights does he spend with his mistress?
A) 25 B) 35 C) 50 D) 65
- Three couples, the Holmeses, the Watsons, and the Moriartys went to shop for Christmas gifts. Each person
bought as many gifts as he or she paid in dollars for each. Each woman spent $147 more than her husband.
If Christine bought 9 less gifts than Ralph Watson and Susan bought 47 gifts less than Oscar Holmes, what is
Jennifer's last name?
A) Holmes B) Watson C) Moriarty D) Who cares?
- Tom, Dick, and Harry were in a room where a murder was committed. One of them is the murderer and the other
two witnessed the crime. Each made a statement to the police as follows:
Tom said, "I am innocent."
Dick said, "Tom and I are both innocent."
Harry said, "Tom and Dick are both innocent."
If one and only one of them is telling the truth, who is the murderer?
A) Tom B) Dick C) Harry D) No unique solution.
- A point P moves on a plane in a manner such that the sum of squares of its distances from two fixed
points remains constant. The curve traced by P is
A) a circle, B) an ellipse with positive eccentricity,
C) a parabola, D) a hyperbola.
- Given a positive integer n, the number of combinations of n is defined to
be the number of ways n can be written as the sum of one or more positive integers which may not be all
different and where different orderings of the same set of integers are counted as different.
What is the formula for the number of combinations of n ?
A) 2(n - 1) B) 2n - 1 C) n! D) (n2 - n + 2)/2
- The angles of elevation of a lock tower from two points 100 yards apart are 30° and 60°. Assuming that the base
of the tower and the two points of observation are in a horizontal line, which of the following cannot be the height
(in yards) of the tower?
A) 25Ö3 B) 50Ö3 C) 50Ö6 D) less than 100.
- If x is the root of the equation x3 - 3x - 6 = 0
and y is the root of the equation y3 - 9y2 + 24y - 12 = 0,
then x + y =
A) 3 - 31/3 B) 2 C) 3 D) 6 - 61/3.
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