MAT!
-ICOUNTS
s
33?
37,
1993-94
I National Competition I
Team Round
DO NOT BEGIN UNTIL YOU ARE
IN STRUCTED TO DO SO
This section of the competition consists of ten problems,
which the team has 20 minutes to complete. Team members
may work together in any way to solve the problems. Team
members may talk during this section of the competition.
This round assumes the use of calculators, and calculations
may also be done on scratch paper, but no other aids are
allowed. All answers must be complete, legible, and
simplified to lowest terms. The team captain must record
“. answers on his/her ownproblem sheet. If the team completes
the problems before time is called, use the remaining time
to check your answers.
State
Team
we Members _......--——--—"""" “"""""""-------... . Captain
Total Correct Scorer’s Initials
MATHCOUNTS is a cooperative project of the National Society of Professional Engineers, the CNA Insurance
Companies, the Cray Research Foundation, the General Motors Foundation, the Intel Foundation, Texas Instruments
Incorporated, the National Council of Teachers of Mathematics, and the National Aeronautics and Space Administration.
®
I
97
1. Given = ab —-
a —
b. Find an integer x such that 1.
2. Given four squares with a common vertex where AB = %AC, 2.
AC =
%AD, and AD lAE. What is the reduced ratio of the
=
smaller shaded area to the larger shaded area? Express your
answer as a common fraction.
3. An equilateral triangle is inscribed in a 2" X 2" square as 3.
shown. How many inches are in the length of a side of the
triangle? Express your answer in decimal form to the nearest
hundredth of an inch. p
4. There are 225 steps from the Woodley Park—Zoo Metro subway 4.
stop to street level in Washington, D.C. If Susan can climb
stairs at a rate of 45 steps per minute, and the escalator travels
at a rate of 75 steps per minute, how many minutes will it take
Susan to climb up the moving escalator? Express your answer
as a mixed number.
5. Four different digits are selected at random from the digits 1 5.
through 9. S is the sum of all possible four-digit numbers that
can be created by using these four digits. F is the greatest
common factor of all such sums. Find F.
6. A geometric solid is created by rotating the segment of
2x -1-3y 6 in the first quadrant around the x—axis. What is the
=
ratio of the surface area to the volume of this solid? Express
your answer as a common fraction in simplest radical form.
Ten regular six-sided
dice are rolled and the product of the top
faces is 25 35. What is the largest possible sum of these faces?
-
If AB = ZAC 3AE 3 4DB and a point x is randomly selected
=
on
AB, what is the probability thatx is between E and C? Give
your answer as ‘a common fraction.
If 31993 = 100k + n, where n < 100 is a nonnegative integer,
what is n ‘.7
10. A plane parallel to the base of a square pyramid divides the 10.
volume into two equal amounts. If the dimensions of the
pyramid are six inches for an edge of the base and six inches for
the height, how many inches are in the distance from the plane
to the base of the pyramid? Express your answer in simplest
radical form.
99
