JV Math Bowl
March 11, 2006
Round I
Sample Question: Given the system of equations
3x  y  7
, find the value of y.
3x  2 y  8
1. Becky runs 12 minutes per mile. How many hours will it take her to run 10
miles?
2. In the accompanying figure, ABFG is a rectangle and BC=CD=DE=EF. Find
area( CAD)
. Write your answer as a fraction in lowest terms.
area( AGF )
B
C
E
D
F
G
A
3. What is the slope of the line represented by the equation 3x  4 y  10 ?
4. The directions George has been given are: “go straight 1 mile, then turn left and
go 2 miles. Then turn right and go 3 miles. Then turn left and go 4 miles. Then
turn right and go 5 miles. Then turn left and go 6 miles. Then turn right and go
7 miles. “ If all turns are 90 degrees, how many miles from his starting point
does he end up? Answer must be simplified.
5. On a standard (non-digital) clock, how many degrees does the minute hand
rotate in one minute?
6. Five line segments are in a plane. What is the maximum number of intersection
points they could determine.
7. In rhombus ABCD, ABC  120 . Find the ratio
o
AC
.
BD
D
A
C
B
8. John is 4 years older than Mary. Four years ago, John was twice as old as Mary.
What will the sum of their ages be in 4 years?
9. Find the point of intersection of the two lines 4 x  3 y  44 and
6 x  2 y  14 .
10. When rolling two standard dice, what is the probability their sum is a multiple
of 3? Answer as a fraction in lowest terms.
11. A convex dodecagon has internal angles whose degree measures add up to what?
12. Find the smallest natural number whose proper factors add up to more than the
number.
Round II
1. What is the x-intercept of the line 6 x  8 y  24 ?
2. Loni went up Windy Mountain at 15 mph and came back down at 45mph.
What was Loni’s average speed (in mph – of course)? Answer as a decimal.
3. The first time a test was offered, twenty students took it and their average score
was 70. The second time, fifty students took it and their average was 63. If
both groups are lumped together, what would their average be?
4. Suppose two standard dice are rolled. What is the probability that their product
is a prime number? Answer as a fraction in lowest terms.
5. A line l has equation y  3x  4 . What is the equation of the line that goes
through (6,8) and is perpendicular to l. Answer must be in y=mx+b form.
6. Find the exact area of the parallelogram. Answer in simplified radical form.
10
45
20
7. Equilateral triangles are attached on the outside of the edges of a regular
hexagon. Each of the equilateral triangles has area 2. What is the total enclosed
area?
8. A mostly empty aquarium has a base that is 20cm by 100cm. If it is being
filled up at a rate of 1000 cubic cm per minute, how fast is the water level rising,
in cm per minute?
9. Eleven points are in a plane and no three of them are collinear. How many
different line segments can be drawn connecting these points?
10. How many different combinations of pennies, nickels, dimes, and quarters can
add up to 31 cents?
11. The retail price of a certain pair of shoes is twice the wholesale price. After
sitting on the shelf for over a year, the manager decides to mark them down to
the wholesale price. This represents a discount of how many percent?
12. A regular decagon has how many diagonals?
13. On a certain website, passwords are three characters long. The characters can
be any letter or digit or the “+”, “*”, “!” and “%” symbols. How many possible
passwords are there?
Round III
1. In the accompanying figure, AB is parallel to CD. Find the degree measure
of CAD .
B
50
A
C
60
20
D
2. A thirteen-sided figure has external angles whose measures add up to what?
3. Which of the following expressions is the largest? Leave your answer as the
corresponding letter:
A  44
4
B  24
8
C  84
2
D  42
8
E  82
4
4. Two positive numbers have product 160. If the first number is increased by
25% and the second is decreased by 25%, their product will now be what?
5. A cup holds 5 green pens, 6 red pens, and 7 blue pens. If Ying grabs three pens
at random, what is the probability that she gets one of each color? Answer as a
fraction in lowest terms.
6. Suppose ABCDE is a regular pentagon. What is the measure, in degrees,
of CAD ?
7. The figure below is bounded by 4 equal line segments of length 2 and 4 quarter
circles of radius 2. What is the enclosed area? Leave your answer in terms of
.
8. Sam travels 30 miles in 5 hours one way and 6 hours on the way back. What
was his average speed in mph? Answer in decimals, rounded off to three
significant figures.
9. Two points are 6 units apart. What is the area of the smallest square whose
vertices include these two points?
20
10. Simplify (1  i ) .
11. How many values of x are there so that we can find a y such that together, x and
y satisfy both 3x  4 y  5 and 16 x  9 y  25 ?
12. Suppose the operation  is defined by xy  4 y  3x . Solve x4  10 .
13. In the following figure, how many parallelograms can be traced:
Round IV
1. Suppose l is a line that goes through (6,8) and is perpendicular to y  3x  6 .
What is the x-intercept of l?
2. In convex hexagon ABCDEF, the measure, in degrees, of the supplements of
angles A,B,C,D are 1,2,3,4 respectively. What is the sum of the measures of
E and F ?
3. At Joe’s, the pizzas come in 3 sizes and you can have any combination of 8
toppings (including none of them) on each half of the pizza. Let N be the
number of different kinds of pizza that can be ordered. What is the prime
factorization of N?
4. How many ways can a 4 by 2 grid be covered with 2 by 1 tiles?
Diagram
5. Suppose all possible pairs of vertices of a regular hexagon are connected with
line segments. How many regions is the hexagon divided into?
6. The government of the island of Solia is divided into five counties: Central,
North, East, South, and West. How many ways can a map of Solia be colored –
assuming we have 4 colors and no two adjacent counties are to be colored the
same color.
Island of Solia
_
7. An isosceles trapezoid T has parallel sides of length 20 and 36, which are 6
units apart. What is the perimeter of T?
8. Consider the line z=2y, in the x=0 plane. If this line is rotated around the z-axis,
a double-cone is formed. The intersection of the double-cone and the plane y=1
forms what type of curve? Answer must be written out and spelled correctly.
9. Six Easter eggs are in a basket. There are two blue, two yellow, and two red. If
three eggs are selected randomly, what is the probability that one of each color
is chosen? Answer as a fraction in lowest terms.
10. Pump A can fill up a barrel in 10 minutes. Pump B takes 15 minutes to fill up
the same barrel. How many minutes will it take to fill up the barrel, if both
pumps are used?
11. What is the smallest prime that is the sum of three distinct prime numbers?
12. Squares are attached to the outside of a regular octagon. The side length of each
square is 3. What is the total enclosed area?